a bucket open at the top is in the form of a frustum of a cone with a capacity of 12308.8 cm cube The radii of the top and bottom of circular ends of the bucket are 20 cm and 12 cm respectively. Find the height of the bucket and also the area of metal sheet used in making it
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SOLUTION :
Given :
Bucket is in the form of the frustum of a cone.
Capacity of Frustum of a cone = volume of the frustum of a cone = 12308.8 cm³
Bigger radius (r1) = 20 cm
Smaller radius (r2) = 12 cm
Volume of Frustum of cone =(⅓ π)h ( r1² + r2² + r1r2)
12308.8 = (π/3)h ( 20×20 + 12×12 + 20×12)
12308.8× 3 = π h( 400 + 144+240)
12308.8×3 = 22/7 h(784)
h = 12308.8×3 ×7 / (22 × 784)
h = 12308.8×3 / (22 × 112)
h = 6,154.4 × 3 / 11× 112
h= 18,463.2/1232 = 14.99
h=15 (approximately)
Height of Frustum of cone (h) = 15 cm
Slant height (l) of a frustum cone =√h² + (r1 - r2)²
l = √15² +(20 - 12)² = √225 + (8)²
l = √225 + 64 = √ 289 = 17
l = 17 cm
Surface area of a frustum cone = πl (r1 + r2)
= π × 17 (20+12)
= 22/7 × 17(32)
= 11968/7 = 1,709.7
= 1709.7 cm²
Cost of making 1 cm² = ₹10
Cost of making 1709.7 cm² = ₹(1709.7 × 10) = ₹ 17097
Cost of making 1709.7 cm² = ₹ 17097
Hence, the height of the bucket is 15 cm and the cost of making the bucket is ₹ 17097.
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