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: AM-GM method
Write $S = \pi r^2 + \frac{125\pi}{r} + \frac{125\pi}{r}$. The product of the three terms is $\pi^3\cdot125^2$, a constant.

Step 2: Minimum
Equality in AM-GM occurs when $\pi r^2 = \frac{125\pi}{r}$, so $r=5$. Then $S = 3\times25\pi = 75\pi$. Option (A).

Final Answer:
75 pi. \[ \boxed{\text{(A)}\ 75\pi} \]