This is one of the important question from the cbse board exam point of view because in 2013 and 2017 cbse board exam this question was asked. Question number 5 from RS Aggarwal book page number 823, exercise 17C, chapter volume and surface area of solid.
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A container made of a metal sheet open at the top is of the form of frustum of cone, whose height is 16 cm and the radii of its lower and upper circular edges are 8 cm and 20cm respectively. Find i] the cost of metal sheet used to make the container if it costs 10 per 100 cm² , ii] the cost of milk at the rate of 35 per litre which can fill it completely.
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Given data
Radius of lower end = r = 8 cm
Radius of upper end = R = 20 cm
Height of container frustum = h = 16 cm
Cost of 100 cm2 metal sheet = Rs 10
So, Cost of 1 cm2 metal sheet = 10/100 = Rs 0.1
Let l be the slant height.
l2 = (R-r)2 + h2
= (20-8)2 + 162
= 256 + 144
= 400
or l= 20cm
Surface area of frustum of the cone = πr2 + π(R + r)l cm2
= π[ (20 + 8 )20 + 82]
= π[560 + 64]
= 624 X ( 22/7 )
= 1961.14 cm2
(i) Find cost of metal sheet used to make the container if it costs Rs. 10 per 100 cm2
Cost of metal sheet per 100 cm2 = Rs. 10
Cost of metal for Rs. 1961.14 cm2 = (1961.14 x10)/100 = Rs. 196.114
(ii) Find the volume of frustum:
Volume of frustum = 1/3 πh(r2 + R2 + rR)
= 1/3 x 22/7 x 16(82 + 202 + 8×20)
= 1/3 x 22/7 x 16(64 + 400 + 160)
= 10345.4208 cm3
= 10.345 liters
Using
1000cm3 = 1 litre
1 cm3 = 1/1000 litre
Cost of 1 liter milk = Rs. 35
Cost of 10.345 liter milk = Rs. 35 x 10.345
= Rs. 363