A closed right circular cylinder of volume 250π cm3 is to be made so…

The correct answer is Option (4) → (A), (C) and (D) only

Let r be the radius and h the height. The volume fixes h:

    \( \displaystyle \pi r^{2}h = 250\pi,\ h = \frac{250}{r^{2}} \)

The total surface area is

    \( \displaystyle S = 2\pi r^{2} + 2\pi rh = 2\pi r^{2} + \frac{500\pi}{r} \)

Setting the derivative equal to zero:

    \( \displaystyle S' = 4\pi r - \frac{500\pi}{r^{2}} = 0 \)

    \( r^{3} = 125,\ r = 5 \)

Since \(S'' = 4\pi + \frac{1000\pi}{r^{3}} \gt 0\), this is the minimum. Then

    \( \displaystyle h = \frac{250}{25} = 10 \)

    \( S = 50\pi + 100\pi = 150\pi \)

So the least surface area needs r = 5 cm and h = 10 cm (height = diameter), and it is \(150\pi\) cm².

Checking the statements

(A) its radius is 5 cm. — Correct. S′ = 0 gives r³ = 125.

(B) its height is equal to its radius. — Incorrect. h = 10 cm while r = 5 cm.

(C) its least total surface area is \(150\pi\) cm². — Correct. S = 2π(25) + 2π(5)(10) = 150π.

(D) its height is equal to the diameter of its base. — Correct. h = 10 = 2 × 5.