Question

Kiran manufacturing a cylinder from a triangular prism of the same height, the triangle base is an equilateral one. How much percentage of material Kiran need to remove to get a cylinder of maximum radius?
1. 30.5%
2. 32.5%
3. 36.5%
4. 39.5%

Answer 1

Correct Answer - Option 4 : 39.5%

Calculation:

Let a be triangle side and h be the height of the prism,

The volume of prism = (√3 a²h)/4

The cylinder base would be the inscribed circle of triangle,

The radius of inscribed circle = a/2√3

Volume of cylinder = π r²h

⇒ πa²h/12

Removed material = ((√3 a2h)/4) - (πa2h/12)

⇒ ((3√3 - π)a²h)/12

Percentage = (((3√3 - π)a2h)/12)/((√3 a2h)/4)) × 100

⇒ 39.5% 

The volume of a prism is the product of base area and height.

The radius of the inscribed circle of an equilateral triangle of side length a is a/2√3.