NCERT Solutions for Class 9 Maths Chapter 13 – Surface Areas and Volumes (Ex 13.1 - Ex 13.5)
Page No: 213
Exercise 13.1
1. A plastic box 1.5 m long, 1.25 m wide and 65 cm deep is to be made. It is opened at the top. Ignoring the thickness of the plastic sheet, determine:
(i) The area of the sheet required for making the box.
(ii) The cost of sheet for it, if a sheet measuring 1m2 costs Rs 20.
Answer
Length of plastic box (l) = 1.5 m
Width of plastic box (b) = 1.25 m
Depth of plastic box (h) = 0.65 m
(i) The area of sheet required to make the box is equal to the surface area of the box excluding the top.
Surface area of the box = Lateral surface area + Area of the base
= 2(l+b)×h + (l×b)
= 2[(1.5 + 1.25)×0.65] + (1.5 × 1.25) m2
= (3.575 + 1.875) m2
= 5.45 m2
The sheet required required to make the box is 5.45 m2
(ii) Cost of 1 m2 of sheet = Rs 20
∴ Cost of 5.45 m2 of sheet = Rs (20 × 5.45) = Rs 109
2. The length, breadth and height of a room are 5m, 4m and 3m respectively. Find the cost of white washing the walls of the room and the ceiling at the rate of ₹7.50 per m2.
Answer
length of the room = 5m
breadth of the room = 4m
height of the room = 3m
Area of four walls including the ceiling = 2(l+b)×h + (l×b)
= 2(5+4)×3 + (5×4) m2
= (54 + 20) m2
= 74 m2 Cost of white washing = ₹7.50 per m2
Total cost = ₹ (74×7.50) = ₹ 555
3. The floor of a rectangular hall has a perimeter 250 m. If the cost of painting the four walls at the rate of ₹10 per m2 is ₹15000, find the height of the hall.
[Hint : Area of the four walls = Lateral surface area.]
Answer
Perimeter of rectangular hall = 2(l + b) = 250 m
Total cost of painting = ₹15000
Rate per m2 = ₹10
Area of four walls = 2(l + b) h m2 = (250×h) m2
A/q,
(250×h)×10 = ₹15000
⇒ 2500×h = ₹15000
⇒ h = 15000/2500 m
⇒ h = 6 m
Thus the height of the hall is 6 m.
4. The paint in a certain container is sufficient to paint an area equal to 9.375 m2. How many bricks of dimensions 22.5 cm×10 cm×7.5 cm can be painted out of this container?
Answer
Volume of paint = 9.375 m2 = 93750 cm2
Dimension of brick = 22.5 cm×10 cm×7.5 cm
Total surface area of a brick = 2(lb + bh + lh) cm2
= 2(22.5×10 + 10×7.5 + 22.5×7.5) cm2
= 2(225 + 75 + 168.75) cm2
= 2×468.75 cm2 = 937.5 cm2
Number of bricks can be painted = 93750/937.5 = 100
5. A cubical box has each edge 10 cm and another cuboidal box is 12.5 cm long, 10 cm wide and 8 cm high.
(i) Which box has the greater lateral surface area and by how much?
(ii) Which box has the smaller total surface area and by how much?
Answer
(i) Lateral surface area of cubical box of edge 10cm = 4×102 cm2 = 400 cm2
Lateral surface area of cuboid box = 2(l+b)×h
= 2×(12.5+10)×8 cm2
= 2×22.5×8 cm2 = 360 cm2
Thus, lateral surface area of the cubical box is greater by (400 – 360) cm2 = 40 cm2
(ii) Total surface area of cubical box of edge 10 cm =6×102cm2=600cm2
Total surface area of cuboidal box = 2(lb + bh + lh)
= 2(12.5×10 + 10×8 + 8×12.5) cm2
= 2(125+80+100) cm2
= (2×305) cm2 = 610 cm2
Thus, total surface area of cubical box is smaller by 10 cm2
6. A small indoor greenhouse (herbarium) is made entirely of glass panes (including base) held together with tape. It is 30 cm long, 25 cm wide and 25 cm high.
(i) What is the area of the glass?
(ii) How much of tape is needed for all the 12 edges?
Answer
(i) Dimensions of greenhouse:
l = 30 cm, b = 25 cm, h = 25 cm
Total surface area of green house = 2(lb + bh + lh)
= 2(30×25 + 25×25 + 25×30) cm2
= 2(750 + 625 + 750) cm2
= 4250 cm2
(ii) Length of the tape needed = 4(l + b + h)
= 4(30 + 25 + 25) cm
= 4×80 cm = 320 cm
7. Shanti Sweets Stall was placing an order for making cardboard boxes for packing their sweets. Two sizes of boxes were required. The bigger of dimensions 25 cm × 20 cm × 5 cm and the smaller of dimensions 15 cm × 12 cm × 5 cm. For all the overlaps, 5% of the total surface area is required extra. If the cost of the cardboard is ₹4 for 1000 cm2 , find the cost of cardboard required for supplying 250 boxes of each kind.
Answer
Dimension of bigger box = 25 cm × 20 cm × 5 cm
Total surface area of bigger box = 2(lb + bh + lh)
= 2(25×20 + 20×5 + 25×5) cm2
Exercise 13.1
1. A plastic box 1.5 m long, 1.25 m wide and 65 cm deep is to be made. It is opened at the top. Ignoring the thickness of the plastic sheet, determine:
(i) The area of the sheet required for making the box.
(ii) The cost of sheet for it, if a sheet measuring 1m2 costs Rs 20.
Answer
Length of plastic box (l) = 1.5 m
Width of plastic box (b) = 1.25 m
Depth of plastic box (h) = 0.65 m
(i) The area of sheet required to make the box is equal to the surface area of the box excluding the top.
Surface area of the box = Lateral surface area + Area of the base
= 2(l+b)×h + (l×b)
= 2[(1.5 + 1.25)×0.65] + (1.5 × 1.25) m2
= (3.575 + 1.875) m2
= 5.45 m2
The sheet required required to make the box is 5.45 m2
(ii) Cost of 1 m2 of sheet = Rs 20
∴ Cost of 5.45 m2 of sheet = Rs (20 × 5.45) = Rs 109
2. The length, breadth and height of a room are 5m, 4m and 3m respectively. Find the cost of white washing the walls of the room and the ceiling at the rate of ₹7.50 per m2.
Answer
length of the room = 5m
breadth of the room = 4m
height of the room = 3m
Area of four walls including the ceiling = 2(l+b)×h + (l×b)
= 2(5+4)×3 + (5×4) m2
= (54 + 20) m2
= 74 m2 Cost of white washing = ₹7.50 per m2
Total cost = ₹ (74×7.50) = ₹ 555
3. The floor of a rectangular hall has a perimeter 250 m. If the cost of painting the four walls at the rate of ₹10 per m2 is ₹15000, find the height of the hall.
[Hint : Area of the four walls = Lateral surface area.]
Answer
Perimeter of rectangular hall = 2(l + b) = 250 m
Total cost of painting = ₹15000
Rate per m2 = ₹10
Area of four walls = 2(l + b) h m2 = (250×h) m2
A/q,
(250×h)×10 = ₹15000
⇒ 2500×h = ₹15000
⇒ h = 15000/2500 m
⇒ h = 6 m
Thus the height of the hall is 6 m.
4. The paint in a certain container is sufficient to paint an area equal to 9.375 m2. How many bricks of dimensions 22.5 cm×10 cm×7.5 cm can be painted out of this container?
Answer
Volume of paint = 9.375 m2 = 93750 cm2
Dimension of brick = 22.5 cm×10 cm×7.5 cm
Total surface area of a brick = 2(lb + bh + lh) cm2
= 2(22.5×10 + 10×7.5 + 22.5×7.5) cm2
= 2(225 + 75 + 168.75) cm2
= 2×468.75 cm2 = 937.5 cm2
Number of bricks can be painted = 93750/937.5 = 100
5. A cubical box has each edge 10 cm and another cuboidal box is 12.5 cm long, 10 cm wide and 8 cm high.
(i) Which box has the greater lateral surface area and by how much?
(ii) Which box has the smaller total surface area and by how much?
Answer
(i) Lateral surface area of cubical box of edge 10cm = 4×102 cm2 = 400 cm2
Lateral surface area of cuboid box = 2(l+b)×h
= 2×(12.5+10)×8 cm2
= 2×22.5×8 cm2 = 360 cm2
Thus, lateral surface area of the cubical box is greater by (400 – 360) cm2 = 40 cm2
(ii) Total surface area of cubical box of edge 10 cm =6×102cm2=600cm2
Total surface area of cuboidal box = 2(lb + bh + lh)
= 2(12.5×10 + 10×8 + 8×12.5) cm2
= 2(125+80+100) cm2
= (2×305) cm2 = 610 cm2
Thus, total surface area of cubical box is smaller by 10 cm2
6. A small indoor greenhouse (herbarium) is made entirely of glass panes (including base) held together with tape. It is 30 cm long, 25 cm wide and 25 cm high.
(i) What is the area of the glass?
(ii) How much of tape is needed for all the 12 edges?
Answer
(i) Dimensions of greenhouse:
l = 30 cm, b = 25 cm, h = 25 cm
Total surface area of green house = 2(lb + bh + lh)
= 2(30×25 + 25×25 + 25×30) cm2
= 2(750 + 625 + 750) cm2
= 4250 cm2
(ii) Length of the tape needed = 4(l + b + h)
= 4(30 + 25 + 25) cm
= 4×80 cm = 320 cm
7. Shanti Sweets Stall was placing an order for making cardboard boxes for packing their sweets. Two sizes of boxes were required. The bigger of dimensions 25 cm × 20 cm × 5 cm and the smaller of dimensions 15 cm × 12 cm × 5 cm. For all the overlaps, 5% of the total surface area is required extra. If the cost of the cardboard is ₹4 for 1000 cm2 , find the cost of cardboard required for supplying 250 boxes of each kind.
Answer
Dimension of bigger box = 25 cm × 20 cm × 5 cm
Total surface area of bigger box = 2(lb + bh + lh)
= 2(25×20 + 20×5 + 25×5) cm2
= 2(500 + 100 + 125) cm2
= 1450 cm2
Dimension of smaller box = 15 cm × 12 cm × 5 cm
Total surface area of smaller box = 2(lb + bh + lh)
= 2(15×12 + 12×5 + 15×5) cm2
= 2(180 + 60 + 75) cm2
= 630 cm2
Total surface area of 250 boxes of each type = 250(1450 + 630) cm2
= 250×2080 cm2 = 520000 cm2
Extra area required = 5/100(1450 + 630) × 250 cm2 = 26000 cm2
Total Cardboard required = 520000 + 26000 cm2 = 546000 cm2
Total cost of cardboard sheet = ₹ (546000 × 4)/1000 = ₹ 2184
8. Parveen wanted to make a temporary shelter for her car, by making a box-like structure with tarpaulin that covers all the four sides and the top of the car (with the front face as a flap which can be rolled up). Assuming that the stitching margins are very small, and therefore negligible, how much tarpaulin would be required to make the shelter of height 2.5 m, with base dimensions 4m × 3m?
Answer
Dimensions of the box- like structure = 4m × 3m × 2.5
Tarpaulin only required for all the four sides and top.
Thus, Tarpaulin required = 2(l+b)×h + lb
= [2(4+3)×2.5 + 4×3] m2
= (35+12) m2
= 47 m2
Page No: 216
Exercise 13.2
1. The curved surface area of a right circular cylinder of height 14 cm is 88 cm2. Find the diameter of the base of the cylinder.
Answer
Let r be the radius of the base and h = 14 cm be the height of the cylinder.
Curved surface area of cylinder = 2πrh = 88 cm2
⇒ 2 × 22/7 × r × 14 = 88
⇒ r = 88/ (2 × 22/7 × 14)
⇒ r = 1 cm
Thus, the diameter of the base = 2r = 2×1 = 2cm
2. It is required to make a closed cylindrical tank of height 1 m and base diameter 140 cm from a metal sheet. How many square metres of the sheet are required for the same?
Answer
Let r be the radius of the base and h be the height of the cylinder.
Base diameter = 140 cm and Height (h) = 1m
Radius of base (r) = 140/2 = 70 cm = 0.7 m
Metal sheet required to make a closed cylindrical tank = 2πr(h + r)
= (2 × 22/7 × 0.7) (1 + 0.7) m2
= (2 × 22 × 0.1 × 1.7) m2
=7.48 m2
3. A metal pipe is 77 cm long. The inner diameter of a cross section is 4 cm, the outer diameter being 4.4 cm (see Fig. 13.11). Find its
(i) inner curved surface area,
(ii) outer curved surface area,
(iii) total surface area.
h = 77 cm
(i) Inner curved surface = 2πrh cm2
= 2 × 22/7 × 2 × 77cm2
= 968 cm2
= 1064.8 cm2
= [968 + 1064.8 + (2 × 22/7) (4.84 - 4)] cm2
= (2032.8 + 44/7 × 0.84) cm2
= (2032.8 + 5.28) cm2 = 2038.08 cm2
Page No: 217
4. The diameter of a roller is 84 cm and its length is 120 cm. It takes 500 complete revolutions to move once over to level a playground. Find the area of the playground in m2.
Answer
Length of the roller (h) = 120 cm = 1.2 m
Radius of the cylinder = 84/2 cm = 42 cm = 0.42 m
Total no. of revolutions = 500
Distance covered by roller in one revolution = Curved surface area = 2πrh
= (2 × 22/7 × 0.42 × 1.2) m2
= 3.168 m2
Area of the playground = (500 × 3.168) m2 = 1584 m2
5. A cylindrical pillar is 50 cm in diameter and 3.5 m in height. Find the cost of painting the curved surface of the pillar at the rate of ₹12.50 per m2.
Answer
Radius of the pillar (r) = 50/2 cm = 25 cm = 0.25 m
Height of the pillar (h) = 3.5 m.
Rate of painting = ₹12.50 per m2
Curved surface = 2πrh
= (2 × 22/7 × 0.25 × 3.5) m2
=5.5 m2
Total cost of painting = (5.5 × 12.5) = ₹68.75
6. Curved surface area of a right circular cylinder is 4.4 m2. If the radius of the base of the cylinder is 0.7 m, find its height.
Answer
Let r be the radius of the base and h be the height of the cylinder.
Curved surface area = 2πrh = 4.4 m2
⇒ 2 × 22/7 × 0.7 × h = 4.4
⇒ h = 4.4/(2 × 22/7 × 0.7) = 1m
⇒ h = 1m
7. The inner diameter of a circular well is 3.5 m. It is 10 m deep. Find
(i) its inner curved surface area,
(ii) the cost of plastering this curved surface at the rate of ₹40 per m2.
Answer
Radius of circular well (r) = 3.5/2 m = 1.75 m
Depth of the well (h) = 10 m
Rate of plastering = ₹40 per m2
(i) Curved surface = 2πrh
= (2 × 22/7 × 1.75 × 10) m2
= 110 m2
(ii) Cost of plastering = ₹(110 × 40) = ₹4400
8. In a hot water heating system, there is a cylindrical pipe of length 28 m and diameter 5 cm. Find the total radiating surface in the system.
Answer
Radius of the pipe (r) = 5/2 cm = 2.5 cm = 0.025 m
Length of the pipe (h) = 28/2 m = 14 m
Total radiating surface = Curved surface area of the pipe = 2πrh
= (2 × 22/7 × 0.025 × 28) m2
= 4.4 m2
9. Find
(i) the lateral or curved surface area of a closed cylindrical petrol storage tank that is 4.2 m in diameter and 4.5 m high.
(ii) how much steel was actually used, if 1/12 of the steel actually used was wasted in making the tank.
Answer
(i) Radius of the tank (r) = 4.2/2 m = 2.1 m
Height of the tank (h) = 4.5 m
Curved surface area = 2πrh m2
= (2 × 22/7 × 2.1 × 4.5) m2
= 59.4 m2
(ii) Total surface area of the tank = 2πr(r + h) m2
= [2 × 22/7 × 2.1 (2.1 + 4.5)] m2
= 87.12 m2
Let x be the actual steel used in making tank.
∴ (1 - 1/12) × x = 87.12
⇒ x = 87.12 × 12/11
⇒ x = 95.04 m2
10. In Fig. 13.12, you see the frame of a lampshade. It is to be covered with a decorative cloth. The frame has a base diameter of 20 cm and height of 30 cm. A margin of 2.5 cm is to be given for folding it over the top and bottom of the frame. Find how much cloth is required for covering the lampshade.
Answer
Radius of the frame (r) = 20/2 cm = 10 cm
Height of the frame (h) = 30 cm + 2×2.5 cm = 35 cm
2.5 cm of margin will be added both side in the height.
Cloth required for covering the lampshade = curved surface area = 2πrh
= (2 × 22/7 × 10 × 35)cm2
= 2200 cm2
11. The students of a Vidyalaya were asked to participate in a competition for making and decorating penholders in the shape of a cylinder with a base, using cardboard. Each penholder was to be of radius 3 cm and height 10.5 cm. The Vidyalaya was to supply the competitors with cardboard. If there were 35 competitors, how much cardboard was required to be bought for the competition?
Answer
Radius of the penholder (r) = 3cm
Height of the penholder (h) = 10.5cm
Cardboard required by 1 competitor = CSA of one penholder + area of the base
= 2πrh + πr2
= [(2 × 22/7 × 3 × 10.5) + 22/7 × 32] cm2
= (198 + 198/7) cm2
= 1584/7 cm2
Cardboard required for 35 competitors = (35 × 1584/7) cm2
= 7920 cm2
Page No: 221
1. Diameter of the base of a cone is 10.5 cm and its slant height is 10 cm. Find its curved surface area.
Answer
Radius (r) = 10.5/2 cm = 5.25 cm
Slant height (l) = 10 cm
Curved surface area of the cone = (πrl) cm2
=(22/7 × 5.25 × 10) cm2
=165 cm2
2. Find the total surface area of a cone, if its slant height is 21 m and diameter of its base is 24 m.
Answer
Radius (r) = 24/2 m = 12 m
Slant height (l) = 21 m
Total surface area of the cone = πr (l + r) m2
= 22/7 × 12 × (21 + 12) m2
= (22/7 × 12 × 33) m2
= 1244.57 m2
3. Curved surface area of a cone is 308 cm2 and its slant height is 14 cm. Find
(i) radius of the base and (ii) total surface area of the cone.
Answer
(i) Curved surface of a cone = 308 cm2
Slant height (l) = 14cm
Let r be the radius of the base
∴ πrℓ = 308
⇒ 22/7 × r × 14 = 308
⇒ r =308/(22/7 × 14) = 7 cm
(ii) TSA of the cone = πr(l + r) cm2
= 22/7 × 7 ×(14 + 7) cm2
= (22 × 21) cm2
= 462 cm2
4. A conical tent is 10 m high and the radius of its base is 24 m. Find
(i) slant height of the tent.
(ii) cost of the canvas required to make the tent, if the cost of 1 m2 canvas is ₹70.
Answer
Radius of the base (r) = 24 m
Height of the conical tent (h) = 10 m
Let l be the slant height of the cone.
∴ l2 = h2 + r2
⇒ l = √h2 + r2
⇒ l = √102 + 242
⇒ l = √100 + 576
⇒ l = 26 m
(ii) Canvas required to make the conical tent = Curved surface of the cone
Cost of 1 m2 canvas = ₹70
∴ πrl = (22/7 × 24 × 26) m2 = 13728/7 m2
∴ Cost of canvas = ₹ 13728/7 × 70 = ₹137280
5. What length of tarpaulin 3m wide will be required to make conical tent of height 8m and base radius 6m? Assume that the extra length of material that will be required for stitching margins and wastage in cutting is approximately 20cm (Use π = 3.14).
Answer
Radius of the base (r) = 6 m
Height of the conical tent (h) = 8 m
Let l be the slant height of the cone.
∴ l = √h2 + r2
⇒ l = √82 + 62
⇒ l = √100
⇒ l = 10 m
CSA of conical tent = πrl
= (3.14 × 6 × 10) m2 = 188.4 m2
Breadth of tarpaulin = 3 m
Let length of tarpaulin sheet required be x.
20 cm will be wasted in cutting.
So, the length will be (x - 0.2 m)
Breadth of tarpaulin = 3 m
Area of sheet = CSA of tent
[(x - 0.2 m) × 3] m = 188.4 m2
⇒ x - 0.2 m = 62.8 m
⇒ x = 63 m
6. The slant height and base diameter of a conical tomb are 25 m and 14 m respectively. Find the cost of white-washing its curved surface at the rate of ₹210 per 100 m2.
Answer
Radius (r) = 14/2 m = 7 m
Slant height tomb (l) = 25 m
Curved surface area = πrl m2
=(227×25×7) m2
=550 m2
Rate of white- washing = ₹210 per 100 m2
Total cost of white-washing the tomb = ₹(550 × 210/100) = ₹1155
7. A joker’s cap is in the form of a right circular cone of base radius 7 cm and height 24 cm. Find the area of the sheet required to make 10 such caps.
Answer
Radius of the cone (r) = 7 cm
Height of the cone (h) = 24 cm
Let l be the slant height
∴ l = √h2 + r2
⇒ l = √242 + 72
⇒ l = √625
⇒ l = 25 m
Sheet required for one cap = Curved surface of the cone
= πrl cm2
= (22/7 × 7 × 25) cm2
= 550 cm2
Sheet required for 10 caps = 550 × 10 cm2 = 5500 cm2
8. A bus stop is barricaded from the remaining part of the road, by using 50 hollow cones made of recycled cardboard. Each cone has a base diameter of 40 cm and height 1 m. If the outer side of each of the cones is to be painted and the cost of painting is ₹12 per m2 , what will be the cost of painting all these cones? (Use π = 3.14 and take √1.04 = 1.02)
Answer
Radius of the cone (r) = 40/2 cm = 20 cm = 0.2 m
Height of the cone (h) = 1 m
Let l be the slant height of a cone.
∴ l = √h2 + r2
⇒ l = √12 + 0.22
⇒ l = √1.04
⇒ l = 1.02 m
Rate of painting = ₹12 per m2
Curved surface of 1 cone = πrl m2
= (3.14 × 0.2 × 1.02) m2
= 0.64056 m2
Curved surface of such 50 cones = (50 × 0.64056) m2
= 32.028 m2
Cost of painting all these cones = ₹(32.028 × 12)
= ₹384.34
= 1450 cm2
Dimension of smaller box = 15 cm × 12 cm × 5 cm
Total surface area of smaller box = 2(lb + bh + lh)
= 2(15×12 + 12×5 + 15×5) cm2
= 2(180 + 60 + 75) cm2
= 630 cm2
Total surface area of 250 boxes of each type = 250(1450 + 630) cm2
= 250×2080 cm2 = 520000 cm2
Extra area required = 5/100(1450 + 630) × 250 cm2 = 26000 cm2
Total Cardboard required = 520000 + 26000 cm2 = 546000 cm2
Total cost of cardboard sheet = ₹ (546000 × 4)/1000 = ₹ 2184
8. Parveen wanted to make a temporary shelter for her car, by making a box-like structure with tarpaulin that covers all the four sides and the top of the car (with the front face as a flap which can be rolled up). Assuming that the stitching margins are very small, and therefore negligible, how much tarpaulin would be required to make the shelter of height 2.5 m, with base dimensions 4m × 3m?
Answer
Dimensions of the box- like structure = 4m × 3m × 2.5
Tarpaulin only required for all the four sides and top.
Thus, Tarpaulin required = 2(l+b)×h + lb
= [2(4+3)×2.5 + 4×3] m2
= (35+12) m2
= 47 m2
Page No: 216
Exercise 13.2
1. The curved surface area of a right circular cylinder of height 14 cm is 88 cm2. Find the diameter of the base of the cylinder.
Answer
Let r be the radius of the base and h = 14 cm be the height of the cylinder.
Curved surface area of cylinder = 2πrh = 88 cm2
⇒ 2 × 22/7 × r × 14 = 88
⇒ r = 88/ (2 × 22/7 × 14)
⇒ r = 1 cm
Thus, the diameter of the base = 2r = 2×1 = 2cm
2. It is required to make a closed cylindrical tank of height 1 m and base diameter 140 cm from a metal sheet. How many square metres of the sheet are required for the same?
Answer
Let r be the radius of the base and h be the height of the cylinder.
Base diameter = 140 cm and Height (h) = 1m
Radius of base (r) = 140/2 = 70 cm = 0.7 m
Metal sheet required to make a closed cylindrical tank = 2πr(h + r)
= (2 × 22/7 × 0.7) (1 + 0.7) m2
= (2 × 22 × 0.1 × 1.7) m2
=7.48 m2
3. A metal pipe is 77 cm long. The inner diameter of a cross section is 4 cm, the outer diameter being 4.4 cm (see Fig. 13.11). Find its
(i) inner curved surface area,
(ii) outer curved surface area,
(iii) total surface area.
Answer
Let R be external radius and r be the internal radius h be the length of the pipe.
R = 4.4/2 cm = 2.2 cm
r = 4/2 cm = 2 cmh = 77 cm
(i) Inner curved surface = 2πrh cm2
= 2 × 22/7 × 2 × 77cm2
= 968 cm2
(ii) Outer curved surface = 2πRh cm2
= 2 × 22/7 × 2.2 × 77 cm2= 1064.8 cm2
(iii) Total surface area of a pipe = Inner curved surface area + outer curved surface area + areas of two bases
= 2πrh + 2πRh + 2π(R2 - r2)= [968 + 1064.8 + (2 × 22/7) (4.84 - 4)] cm2
= (2032.8 + 44/7 × 0.84) cm2
= (2032.8 + 5.28) cm2 = 2038.08 cm2
Page No: 217
4. The diameter of a roller is 84 cm and its length is 120 cm. It takes 500 complete revolutions to move once over to level a playground. Find the area of the playground in m2.
Answer
Length of the roller (h) = 120 cm = 1.2 m
Radius of the cylinder = 84/2 cm = 42 cm = 0.42 m
Total no. of revolutions = 500
Distance covered by roller in one revolution = Curved surface area = 2πrh
= (2 × 22/7 × 0.42 × 1.2) m2
= 3.168 m2
Area of the playground = (500 × 3.168) m2 = 1584 m2
5. A cylindrical pillar is 50 cm in diameter and 3.5 m in height. Find the cost of painting the curved surface of the pillar at the rate of ₹12.50 per m2.
Answer
Radius of the pillar (r) = 50/2 cm = 25 cm = 0.25 m
Height of the pillar (h) = 3.5 m.
Rate of painting = ₹12.50 per m2
Curved surface = 2πrh
= (2 × 22/7 × 0.25 × 3.5) m2
=5.5 m2
Total cost of painting = (5.5 × 12.5) = ₹68.75
6. Curved surface area of a right circular cylinder is 4.4 m2. If the radius of the base of the cylinder is 0.7 m, find its height.
Answer
Let r be the radius of the base and h be the height of the cylinder.
Curved surface area = 2πrh = 4.4 m2
⇒ 2 × 22/7 × 0.7 × h = 4.4
⇒ h = 4.4/(2 × 22/7 × 0.7) = 1m
⇒ h = 1m
7. The inner diameter of a circular well is 3.5 m. It is 10 m deep. Find
(i) its inner curved surface area,
(ii) the cost of plastering this curved surface at the rate of ₹40 per m2.
Answer
Radius of circular well (r) = 3.5/2 m = 1.75 m
Depth of the well (h) = 10 m
Rate of plastering = ₹40 per m2
(i) Curved surface = 2πrh
= (2 × 22/7 × 1.75 × 10) m2
= 110 m2
(ii) Cost of plastering = ₹(110 × 40) = ₹4400
8. In a hot water heating system, there is a cylindrical pipe of length 28 m and diameter 5 cm. Find the total radiating surface in the system.
Answer
Radius of the pipe (r) = 5/2 cm = 2.5 cm = 0.025 m
Length of the pipe (h) = 28/2 m = 14 m
Total radiating surface = Curved surface area of the pipe = 2πrh
= (2 × 22/7 × 0.025 × 28) m2
= 4.4 m2
9. Find
(i) the lateral or curved surface area of a closed cylindrical petrol storage tank that is 4.2 m in diameter and 4.5 m high.
(ii) how much steel was actually used, if 1/12 of the steel actually used was wasted in making the tank.
Answer
(i) Radius of the tank (r) = 4.2/2 m = 2.1 m
Height of the tank (h) = 4.5 m
Curved surface area = 2πrh m2
= (2 × 22/7 × 2.1 × 4.5) m2
= 59.4 m2
(ii) Total surface area of the tank = 2πr(r + h) m2
= [2 × 22/7 × 2.1 (2.1 + 4.5)] m2
= 87.12 m2
Let x be the actual steel used in making tank.
∴ (1 - 1/12) × x = 87.12
⇒ x = 87.12 × 12/11
⇒ x = 95.04 m2
10. In Fig. 13.12, you see the frame of a lampshade. It is to be covered with a decorative cloth. The frame has a base diameter of 20 cm and height of 30 cm. A margin of 2.5 cm is to be given for folding it over the top and bottom of the frame. Find how much cloth is required for covering the lampshade.
Answer
Radius of the frame (r) = 20/2 cm = 10 cm
Height of the frame (h) = 30 cm + 2×2.5 cm = 35 cm
2.5 cm of margin will be added both side in the height.
Cloth required for covering the lampshade = curved surface area = 2πrh
= (2 × 22/7 × 10 × 35)cm2
= 2200 cm2
11. The students of a Vidyalaya were asked to participate in a competition for making and decorating penholders in the shape of a cylinder with a base, using cardboard. Each penholder was to be of radius 3 cm and height 10.5 cm. The Vidyalaya was to supply the competitors with cardboard. If there were 35 competitors, how much cardboard was required to be bought for the competition?
Answer
Radius of the penholder (r) = 3cm
Height of the penholder (h) = 10.5cm
Cardboard required by 1 competitor = CSA of one penholder + area of the base
= 2πrh + πr2
= [(2 × 22/7 × 3 × 10.5) + 22/7 × 32] cm2
= (198 + 198/7) cm2
= 1584/7 cm2
Cardboard required for 35 competitors = (35 × 1584/7) cm2
= 7920 cm2
Page No: 221
1. Diameter of the base of a cone is 10.5 cm and its slant height is 10 cm. Find its curved surface area.
Answer
Radius (r) = 10.5/2 cm = 5.25 cm
Slant height (l) = 10 cm
Curved surface area of the cone = (πrl) cm2
=(22/7 × 5.25 × 10) cm2
=165 cm2
2. Find the total surface area of a cone, if its slant height is 21 m and diameter of its base is 24 m.
Answer
Radius (r) = 24/2 m = 12 m
Slant height (l) = 21 m
Total surface area of the cone = πr (l + r) m2
= 22/7 × 12 × (21 + 12) m2
= (22/7 × 12 × 33) m2
= 1244.57 m2
3. Curved surface area of a cone is 308 cm2 and its slant height is 14 cm. Find
(i) radius of the base and (ii) total surface area of the cone.
Answer
(i) Curved surface of a cone = 308 cm2
Slant height (l) = 14cm
Let r be the radius of the base
∴ πrℓ = 308
⇒ 22/7 × r × 14 = 308
⇒ r =308/(22/7 × 14) = 7 cm
(ii) TSA of the cone = πr(l + r) cm2
= 22/7 × 7 ×(14 + 7) cm2
= (22 × 21) cm2
= 462 cm2
4. A conical tent is 10 m high and the radius of its base is 24 m. Find
(i) slant height of the tent.
(ii) cost of the canvas required to make the tent, if the cost of 1 m2 canvas is ₹70.
Answer
Radius of the base (r) = 24 m
Height of the conical tent (h) = 10 m
Let l be the slant height of the cone.
∴ l2 = h2 + r2
⇒ l = √h2 + r2
⇒ l = √102 + 242
⇒ l = √100 + 576
⇒ l = 26 m
(ii) Canvas required to make the conical tent = Curved surface of the cone
Cost of 1 m2 canvas = ₹70
∴ πrl = (22/7 × 24 × 26) m2 = 13728/7 m2
∴ Cost of canvas = ₹ 13728/7 × 70 = ₹137280
5. What length of tarpaulin 3m wide will be required to make conical tent of height 8m and base radius 6m? Assume that the extra length of material that will be required for stitching margins and wastage in cutting is approximately 20cm (Use π = 3.14).
Answer
Radius of the base (r) = 6 m
Height of the conical tent (h) = 8 m
Let l be the slant height of the cone.
∴ l = √h2 + r2
⇒ l = √82 + 62
⇒ l = √100
⇒ l = 10 m
CSA of conical tent = πrl
= (3.14 × 6 × 10) m2 = 188.4 m2
Breadth of tarpaulin = 3 m
Let length of tarpaulin sheet required be x.
20 cm will be wasted in cutting.
So, the length will be (x - 0.2 m)
Breadth of tarpaulin = 3 m
Area of sheet = CSA of tent
[(x - 0.2 m) × 3] m = 188.4 m2
⇒ x - 0.2 m = 62.8 m
⇒ x = 63 m
6. The slant height and base diameter of a conical tomb are 25 m and 14 m respectively. Find the cost of white-washing its curved surface at the rate of ₹210 per 100 m2.
Answer
Radius (r) = 14/2 m = 7 m
Slant height tomb (l) = 25 m
Curved surface area = πrl m2
=(227×25×7) m2
=550 m2
Rate of white- washing = ₹210 per 100 m2
Total cost of white-washing the tomb = ₹(550 × 210/100) = ₹1155
7. A joker’s cap is in the form of a right circular cone of base radius 7 cm and height 24 cm. Find the area of the sheet required to make 10 such caps.
Answer
Radius of the cone (r) = 7 cm
Height of the cone (h) = 24 cm
Let l be the slant height
∴ l = √h2 + r2
⇒ l = √242 + 72
⇒ l = √625
⇒ l = 25 m
Sheet required for one cap = Curved surface of the cone
= πrl cm2
= (22/7 × 7 × 25) cm2
= 550 cm2
Sheet required for 10 caps = 550 × 10 cm2 = 5500 cm2
8. A bus stop is barricaded from the remaining part of the road, by using 50 hollow cones made of recycled cardboard. Each cone has a base diameter of 40 cm and height 1 m. If the outer side of each of the cones is to be painted and the cost of painting is ₹12 per m2 , what will be the cost of painting all these cones? (Use π = 3.14 and take √1.04 = 1.02)
Answer
Radius of the cone (r) = 40/2 cm = 20 cm = 0.2 m
Height of the cone (h) = 1 m
Let l be the slant height of a cone.
∴ l = √h2 + r2
⇒ l = √12 + 0.22
⇒ l = √1.04
⇒ l = 1.02 m
Rate of painting = ₹12 per m2
Curved surface of 1 cone = πrl m2
= (3.14 × 0.2 × 1.02) m2
= 0.64056 m2
Curved surface of such 50 cones = (50 × 0.64056) m2
= 32.028 m2
Cost of painting all these cones = ₹(32.028 × 12)
= ₹384.34
Page No: 225
Exercise 13.4
1. Find the surface area of a sphere of radius:
(i) 10.5 cm (ii) 5.6 cm (iii) 14 cm
Answer
(i) Radius of the sphere (r) = 10.5 cm
Surface area = 4πr2
= (4 × 22/7 × 10.5 × 10.5) cm2
= 1386 cm2
(ii) Radius of the sphere (r) = 5.6 cm
Surface area = 4πr2
= (4 × 22/7 × 5.6 × 5.6) cm2
= 394.24 cm2
(iii) Radius of the sphere (r) = 14 cm
Surface area = 4πr2
= (4 × 22/7 × 14 × 14) cm2
= 2464 cm2
2. Find the surface area of a sphere of diameter:
(i) 14 cm (ii) 21 cm (iii) 3.5 m
Answer
(i) r = 14/2 cm = 7cm
Surface area = 4πr2
= (4 × 22/7 × 7 × 7)cm2
=616cm2
(ii) r = 21/2 cm = 10.5 cm
Surface area = 4πr2
= (4 × 22/7 × 10.5 × 10.5) cm2
= 1386 cm2
(iii) r = 3.5/2 m = 1.75 m
Surface area = 4πr2
= (4 × 22/7 × 1.75 × 1.75) m2
= 38.5 m2
3. Find the total surface area of a hemisphere of radius 10 cm. (Use π = 3.14)
Answer
r = 10 cm
Total surface area of hemisphere = 3πr2
= (3 × 3.14 × 10 ×10) cm2
= 942 cm2
4. The radius of a spherical balloon increases from 7cm to 14cm as air is being pumped into it. Find the ratio of surface areas of the balloon in the two cases.
Answer
Let r be the initial radius and R be the increased radius of balloons.
r = 7cm and R = 14cm
Ratio of the surface area =4πr2/4πR2
= r2/R2
= (7×7)/(14×14) = 1/4
Thus, the ratio of surface areas = 1 : 4
5. A hemispherical bowl made of brass has inner diameter 10.5 cm. Find the cost of tin-plating it on the inside at the rate of ₹16 per 100 cm2.
Answer
Radius of the bowl (r) = 10.5/2 cm = 5.25 cm
Curved surface area of the hemispherical bowl = 2πr2
= (2 × 22/7 × 5.25 × 5.25) cm2
= 173.25 cm2
Rate of tin - plating is = ₹16 per 100 cm2
Therefor, cost of 1 cm2 = ₹16/100
Total cost of tin-plating the hemisphere bowl = 173.25 × 16/100
Exercise 13.4
1. Find the surface area of a sphere of radius:
(i) 10.5 cm (ii) 5.6 cm (iii) 14 cm
Answer
(i) Radius of the sphere (r) = 10.5 cm
Surface area = 4πr2
= (4 × 22/7 × 10.5 × 10.5) cm2
= 1386 cm2
(ii) Radius of the sphere (r) = 5.6 cm
Surface area = 4πr2
= (4 × 22/7 × 5.6 × 5.6) cm2
= 394.24 cm2
(iii) Radius of the sphere (r) = 14 cm
Surface area = 4πr2
= (4 × 22/7 × 14 × 14) cm2
= 2464 cm2
2. Find the surface area of a sphere of diameter:
(i) 14 cm (ii) 21 cm (iii) 3.5 m
Answer
(i) r = 14/2 cm = 7cm
Surface area = 4πr2
= (4 × 22/7 × 7 × 7)cm2
=616cm2
(ii) r = 21/2 cm = 10.5 cm
Surface area = 4πr2
= (4 × 22/7 × 10.5 × 10.5) cm2
= 1386 cm2
(iii) r = 3.5/2 m = 1.75 m
Surface area = 4πr2
= (4 × 22/7 × 1.75 × 1.75) m2
= 38.5 m2
3. Find the total surface area of a hemisphere of radius 10 cm. (Use π = 3.14)
Answer
r = 10 cm
Total surface area of hemisphere = 3πr2
= (3 × 3.14 × 10 ×10) cm2
= 942 cm2
4. The radius of a spherical balloon increases from 7cm to 14cm as air is being pumped into it. Find the ratio of surface areas of the balloon in the two cases.
Answer
Let r be the initial radius and R be the increased radius of balloons.
r = 7cm and R = 14cm
Ratio of the surface area =4πr2/4πR2
= r2/R2
= (7×7)/(14×14) = 1/4
Thus, the ratio of surface areas = 1 : 4
5. A hemispherical bowl made of brass has inner diameter 10.5 cm. Find the cost of tin-plating it on the inside at the rate of ₹16 per 100 cm2.
Answer
Radius of the bowl (r) = 10.5/2 cm = 5.25 cm
Curved surface area of the hemispherical bowl = 2πr2
= (2 × 22/7 × 5.25 × 5.25) cm2
= 173.25 cm2
Rate of tin - plating is = ₹16 per 100 cm2
Therefor, cost of 1 cm2 = ₹16/100
Total cost of tin-plating the hemisphere bowl = 173.25 × 16/100
= ₹27.72
6. Find the radius of a sphere whose surface area is 154 cm2.
Answer
Let r be the radius of the sphere.
Surface area = 154 cm2
⇒ 4πr2 = 154
⇒ 4 × 22/7 × r2 = 154
⇒ r2 = 154/(4 × 22/7)
⇒ r2 = 49/4
⇒ r = 7/2 = 3.5 cm
7. The diameter of the moon is approximately one fourth of the diameter of the earth. Find the ratio of their surface areas.
Answer
Let the diameter of earth be r and that of the moon will be r/4
Radius of the earth = r/2
Radius of the moon = r/8
Ratio of their surface area = 4π(r/8)2/4π(r/2)2
= (1/64)/(1/4)
= 4/64 = 1/16
Thus, the ratio of their surface areas is 1:16
8. A hemispherical bowl is made of steel, 0.25 cm thick. The inner radius of the bowl is 5 cm. Find the outer curved surface area of the bowl.
Answer
Inner radius of the bowl (r) = 5 cm
Thickness of the steel = 0.25 cm
∴ outer radius (R) = (r + 0.25) cm
= (5 + 0.25) cm = 5.25 cm
Outer curved surface = 2πR2
= (2 × 22/7 × 5.25 × 5.25) cm2
= 173.25 cm2
9. A right circular cylinder just encloses a sphere of radius r (see Fig. 13.22). Find
(i) surface area of the sphere,
(ii) curved surface area of the cylinder,
(iii) ratio of the areas obtained in (i) and (ii).
Answer
(i) The surface area of the sphere with raius r = 4πr2
(ii) The right circular cylinder just encloses a sphere of radius r.
∴ the radius of the cylinder = r and its height = 2r
∴ Curved surface of cylinder =2πrh
= 2π × r × 2r
= 4πr2
(iii) Ratio of the areas = 4πr2:4πr2 = 1:1
Page No: 228
Exercise 13.5
1. A matchbox measures 4cm × 2.5cm × 1.5cm. What will be the volume of a packet containing 12 such boxes?
Answer
Dimension of matchbox = 4cm × 2.5cm × 1.5cm
l = 4 cm, b = 2.5 cm and h = 1.5 cm
Volume of one matchbox = (l × b × h)
= (4 × 2.5 × 1.5) cm3 = 15 cm3
Volume of a packet containing 12 such boxes = (12 × 15) cm3 = 180 cm3
2. A cuboidal water tank is 6 m long, 5 m wide and 4.5 m deep. How many litres of water can it hold? (1m3 = 1000 l)
Answer
Dimensions of water tank = 6m × 5m × 4.5m
l = 6m , b = 5m and h = 4.5m
Therefore Volume of the tank =ℓbh m3
=(6×5×4.5)m3=135 m3
Therefore , the tank can hold = 135 × 1000 litres [Since 1m3=1000litres]
= 135000 litres of water.
3. A cuboidal vessel is 10 m long and 8 m wide. How high must it be made to hold 380 cubic metres of a liquid?
Answer
Length = 10 m , Breadth = 8 m and Volume = 380 m3
Volume of cuboid = Length × Breadth × Height
⇒ Height = Volume of cuboid/(Length × Breadth)
= 380/(10×8) m
= 4.75m
4. Find the cost of digging a cuboidal pit 8 m long, 6 m broad and 3 m deep at the rate of ₹30 per m3.
Answer
l = 8 m, b = 6 m and h = 3 m
Volume of the pit = lbh m3
= (8×6×3) m3
= 144 m3
Rate of digging = ₹30 per m3
Total cost of digging the pit = ₹(144 × 30) = ₹4320
5. The capacity of a cuboidal tank is 50000 litres of water. Find the breadth of the tank, if its length and depth are respectively 2.5 m and 10 m.
Answer
length = 2.5 m, depth = 10 m and volume = 50000 litres
1m3 = 1000 litres
∴ 50000 litres = 50000/1000 m3 = 50 m3
Breadth = Volume of cuboid/(Length×Depth)
= 50/(2.5×10) m
= 2 m
6. A village, having a population of 4000, requires 150 litres of water per head per day. It has a tank measuring 20m × 15m × 6m. For how many days will the water of this tank last?
Answer
Dimension of tank = 20m × 15m × 6m
l = 20 m , b = 15 m and h = 6 m
Capacity of the tank = lbh m3
= (20×15×6) m3
= 1800 m3
Water requirement per person per day =150 litres
Water required for 4000 person per day = (4000×150) l
= (4000×150)/1000
= 600 m3
Number of days the water will last = Capacity of tank Total water required per day
=(1800/600) = 3
The water will last for 3 days.
7. A godown measures 40m × 25m × 15m. Find the maximum number of wooden crates each measuring 1.5m × 1.25m × 0.5m that can be stored in the godown.
Answer
Dimension of godown = 40 m × 25 m × 15 m
Volume of the godown = (40 × 25 × 15) m3 = 10000 m3
Dimension of crates = 1.5m × 1.25m × 0.5m
Volume of 1 crates = (1.5 × 1.25 × 0.5) m3 = 0.9375 m3
Number of crates that can be stored =Volume of the godown/Volume of 1 crate
= 10000/0.9375 = 10666.66 = 10666
8. A solid cube of side 12 cm is cut into eight cubes of equal volume. What will be the side of the new cube? Also, find the ratio between their surface areas.
Answer
Edge of the cube = 12 cm.
Volume of the cube = (edge)3 cm3
= (12 × 12 × 12) cm3
= 1728 cm3
Number of smaller cube = 8
Volume of the 1 smaller cube =1728/8 cm3 = 216 cm3
Side of the smaller cube = a
a3 = 216
⇒ a = 6 cm
Surface area of the cube = 6 (side)2
Ratio of their surface area = (6 × 12 × 12)/(6 × 6 × 6)
= 4/1 = 4:1
9. A river 3 m deep and 40 m wide is flowing at the rate of 2 km per hour. How much water will fall into the sea in a minute?
Answer
Depth of river (h) = 3 m
Width of river (b) = 40 m
Rate of flow of water (l) = 2 km per hour = (2000/60) m per minute
= 100/3 m per minute
Volume of water flowing into the sea in a minute = lbh m3
= (100/3 × 40 × 3) m3
= 4000 m3
6. Find the radius of a sphere whose surface area is 154 cm2.
Answer
Let r be the radius of the sphere.
Surface area = 154 cm2
⇒ 4πr2 = 154
⇒ 4 × 22/7 × r2 = 154
⇒ r2 = 154/(4 × 22/7)
⇒ r2 = 49/4
⇒ r = 7/2 = 3.5 cm
7. The diameter of the moon is approximately one fourth of the diameter of the earth. Find the ratio of their surface areas.
Answer
Let the diameter of earth be r and that of the moon will be r/4
Radius of the earth = r/2
Radius of the moon = r/8
Ratio of their surface area = 4π(r/8)2/4π(r/2)2
= (1/64)/(1/4)
= 4/64 = 1/16
Thus, the ratio of their surface areas is 1:16
8. A hemispherical bowl is made of steel, 0.25 cm thick. The inner radius of the bowl is 5 cm. Find the outer curved surface area of the bowl.
Answer
Inner radius of the bowl (r) = 5 cm
Thickness of the steel = 0.25 cm
∴ outer radius (R) = (r + 0.25) cm
= (5 + 0.25) cm = 5.25 cm
Outer curved surface = 2πR2
= (2 × 22/7 × 5.25 × 5.25) cm2
= 173.25 cm2
9. A right circular cylinder just encloses a sphere of radius r (see Fig. 13.22). Find
(i) surface area of the sphere,
(ii) curved surface area of the cylinder,
(iii) ratio of the areas obtained in (i) and (ii).
(i) The surface area of the sphere with raius r = 4πr2
(ii) The right circular cylinder just encloses a sphere of radius r.
∴ the radius of the cylinder = r and its height = 2r
∴ Curved surface of cylinder =2πrh
= 2π × r × 2r
= 4πr2
(iii) Ratio of the areas = 4πr2:4πr2 = 1:1
Page No: 228
Exercise 13.5
1. A matchbox measures 4cm × 2.5cm × 1.5cm. What will be the volume of a packet containing 12 such boxes?
Answer
Dimension of matchbox = 4cm × 2.5cm × 1.5cm
l = 4 cm, b = 2.5 cm and h = 1.5 cm
Volume of one matchbox = (l × b × h)
= (4 × 2.5 × 1.5) cm3 = 15 cm3
Volume of a packet containing 12 such boxes = (12 × 15) cm3 = 180 cm3
2. A cuboidal water tank is 6 m long, 5 m wide and 4.5 m deep. How many litres of water can it hold? (1m3 = 1000 l)
Answer
Dimensions of water tank = 6m × 5m × 4.5m
l = 6m , b = 5m and h = 4.5m
Therefore Volume of the tank =ℓbh m3
=(6×5×4.5)m3=135 m3
Therefore , the tank can hold = 135 × 1000 litres [Since 1m3=1000litres]
= 135000 litres of water.
3. A cuboidal vessel is 10 m long and 8 m wide. How high must it be made to hold 380 cubic metres of a liquid?
Answer
Length = 10 m , Breadth = 8 m and Volume = 380 m3
Volume of cuboid = Length × Breadth × Height
⇒ Height = Volume of cuboid/(Length × Breadth)
= 380/(10×8) m
= 4.75m
4. Find the cost of digging a cuboidal pit 8 m long, 6 m broad and 3 m deep at the rate of ₹30 per m3.
Answer
l = 8 m, b = 6 m and h = 3 m
Volume of the pit = lbh m3
= (8×6×3) m3
= 144 m3
Rate of digging = ₹30 per m3
Total cost of digging the pit = ₹(144 × 30) = ₹4320
5. The capacity of a cuboidal tank is 50000 litres of water. Find the breadth of the tank, if its length and depth are respectively 2.5 m and 10 m.
Answer
length = 2.5 m, depth = 10 m and volume = 50000 litres
1m3 = 1000 litres
∴ 50000 litres = 50000/1000 m3 = 50 m3
Breadth = Volume of cuboid/(Length×Depth)
= 50/(2.5×10) m
= 2 m
6. A village, having a population of 4000, requires 150 litres of water per head per day. It has a tank measuring 20m × 15m × 6m. For how many days will the water of this tank last?
Answer
Dimension of tank = 20m × 15m × 6m
l = 20 m , b = 15 m and h = 6 m
Capacity of the tank = lbh m3
= (20×15×6) m3
= 1800 m3
Water requirement per person per day =150 litres
Water required for 4000 person per day = (4000×150) l
= (4000×150)/1000
= 600 m3
Number of days the water will last = Capacity of tank Total water required per day
=(1800/600) = 3
The water will last for 3 days.
7. A godown measures 40m × 25m × 15m. Find the maximum number of wooden crates each measuring 1.5m × 1.25m × 0.5m that can be stored in the godown.
Answer
Dimension of godown = 40 m × 25 m × 15 m
Volume of the godown = (40 × 25 × 15) m3 = 10000 m3
Dimension of crates = 1.5m × 1.25m × 0.5m
Volume of 1 crates = (1.5 × 1.25 × 0.5) m3 = 0.9375 m3
Number of crates that can be stored =Volume of the godown/Volume of 1 crate
= 10000/0.9375 = 10666.66 = 10666
8. A solid cube of side 12 cm is cut into eight cubes of equal volume. What will be the side of the new cube? Also, find the ratio between their surface areas.
Answer
Edge of the cube = 12 cm.
Volume of the cube = (edge)3 cm3
= (12 × 12 × 12) cm3
= 1728 cm3
Number of smaller cube = 8
Volume of the 1 smaller cube =1728/8 cm3 = 216 cm3
Side of the smaller cube = a
a3 = 216
⇒ a = 6 cm
Surface area of the cube = 6 (side)2
Ratio of their surface area = (6 × 12 × 12)/(6 × 6 × 6)
= 4/1 = 4:1
9. A river 3 m deep and 40 m wide is flowing at the rate of 2 km per hour. How much water will fall into the sea in a minute?
Answer
Depth of river (h) = 3 m
Width of river (b) = 40 m
Rate of flow of water (l) = 2 km per hour = (2000/60) m per minute
= 100/3 m per minute
Volume of water flowing into the sea in a minute = lbh m3
= (100/3 × 40 × 3) m3
= 4000 m3
