Question

A cylindrical tank without a top lid is being manufactured to hold a fixed volume of \(125π\) cubic cm. The minimum surface area required to construct this tank is ............square cm

• A. \(75π\)
• B. \(50π\)
• C. \(25π\)
• D. \(125π\)

Hint:

Express height from the volume, write surface area in radius, and minimise.

Answer 1

The Correct Option is A

Step 1: Setup
Volume: \(\pi r^2h = 125\pi\), so \(h = \frac{125}{r^2}\). The tank has no lid, so surface area \(S = \pi r^2 + 2\pi rh\).

Step 2: Single variable
\(S = \pi r^2 + \frac{250\pi}{r}\).

Step 3: Minimise
\(S' = 2\pi r - \frac{250\pi}{r^2} = 0\) gives \(r^3 = 125\), so \(r = 5\). \(S'' = 2\pi + \frac{500\pi}{r^3} > 0\), so this is a minimum.

Step 4: Value
\(S = 25\pi + 50\pi = 75\pi\). Option (A).

Final Answer:
The minimum surface area is 75 pi square cm. \[ \boxed{\text{(A)}\ 75\pi} \]