Question

For the least amount of canvas required for a conical tent of given capacity, find the ratio of its height to the radius of its base.

a) 2:1
b) 1:2
c) 3:1
d) 1:3

Answer 1

The ratio of its height to the radius of its base that minimizes the required canvas is 2:1.

How can you find the ratio of its height to the radius of its base?

The volume of a cone is directly proportional to the product of its base area and height: Volume = (1/3)πr²h.

Therefore, to minimize the amount of canvas needed (while maintaining the same volume), we need to minimize the surface area of the conical tent while keeping its volume constant.

Surface area of a cone: S = πr² + πrl (sum of base area and lateral surface area).

We can consider r and h as variables and apply the principle of Lagrange multipliers to find the minimum value of S while keeping the volume V constant.

h/r = 2

Therefore, the ratio of height to base radius should be 2:1 for the minimum canvas requirement.

Hence, the correct answer is a) 2:1.