Question

If $l+r=12$ here $l$ is length of cylinder and $r$ is radius of cylinder then find maximum value of volume of cylinder

Answer 1

$256\pi$

Answer 2

The volume of a cylinder is $V = \pi r^2 l$. Given the constraint $l+r=12$, we can write $l = 12-r$. Substituting this into the volume formula gives $V(r) = \pi r^2 (12-r) = \pi (12r^2 - r^3)$. The domain for $r$ is $[0, 12]$. To find the maximum volume, we find the derivative of $V(r)$ with respect to $r$: $\frac{dV}{dr} = \pi (24r - 3r^2) = 3\pi r(8-r)$. Setting $\frac{dV}{dr} = 0$ gives critical points $r=0$ and $r=8$. We evaluate $V(r)$ at the critical points and the endpoints of the domain $[0, 12]$: $V(0)=0$, $V(8) = \pi (8^2)(12-8) = \pi (64)(4) = 256\pi$, and $V(12) = \pi (12^2)(12-12) = 0$. Comparing these values, the maximum volume is $256\pi$.