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(A) Let the radius of the base be $r$ and the height be $h$. The volume $V = \pi r^2 h = 125\pi$, so $h = \frac{125}{r^2}$.The surface area $S$ of a c

Vedclass · Accepted Answer

$75$

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(A) Let the radius of the base be $r$ and the height be $h$. The volume $V = \pi r^2 h = 125\pi$, so $h = \frac{125}{r^2}$.The surface area $S$ of a cylinder without a top lid is $S = \pi r^2 + 2\pi rh$.Substitute $h$: $S = \pi r^2 + 2\pi r \left( \frac{125}{r^2} ight) = \pi r^2 + \frac{250\pi}{r}$.To find the minimum, differentiate with respect to $r$: $\frac{dS}{dr} = 2\pi r - \frac{250\pi}{r^2}$.Set $\frac{dS}{dr} = 0$: $2\pi r = \frac{250\pi}{r^2} \implies r^3 = 125 \implies r = 5 	ext{ cm}$.Substitute $r = 5$ into $S$: $S = \pi(5)^2 + \frac{250\pi}{5} = 25\pi + 50\pi = 75\pi 	ext{ cm}^2$.

$A$ cylindrical tank without a top lid is being manufactured to hold a fixed volume of $125\pi \text{ cm}^3$. The minimum surface area required to construct this tank is ............ $\text{cm}^2$. (in $\pi$)

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