If the surface area of a right circular cylinder is given then prove that its height is equal to thediameter of its base when the volume is maximum

Sure, let's prove this step by step.

1. Let's denote the radius of the base of the cylinder as r and the height of the cylinder as h. The surface area (A) of a right circular cylinder is given by the formula A = 2πr(r + h).

2. The volume (V) of a right circular cylinder is given by the formula V = πr²h.

3. We are given that the surface area is constant. So, we can express h in terms of r from the surface area formula: h = A/2πr - r.

4. Substitute h in the volume formula: V = πr²(A/2πr - r) = r(A/2 - r²).

5. To find the maximum volume, we take the derivative of V with respect to r and set it equal to zero: dV/dr = A/2 - 3r² = 0.

6. Solving for r, we get r = sqrt(A/6π).

7. Substituting r back into the equation for h, we get h = A/2πsqrt(A/6π) - sqrt(A/6π) = sqrt(A/6π).

8. Therefore, h = 2r, which means the height is equal to the diameter of the base when the volume is maximum. This completes the proof.