Question

The maximum volume (in cubic units) of a right circular cylinder for which the sum of the square of its height and the square of its radius of base \(48\), is

• A. \(48\pi\)
• B. \(64\pi\)
• C. \(128\pi\)
• D. \(16\pi\)

Hint:

Write the volume in terms of height alone using the given constraint and set its derivative to zero.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
Maximise \(V=\pi r^{2}h\) subject to \(r^{2}+h^{2}=48\).

Step 2: Use the constraint
\(r^{2}=48-h^{2}\), so \(V(h)=\pi(48h-h^{3})\).

Step 3: Find the critical point
\(\frac{dV}{dh}=\pi(48-3h^{2})=0\) gives \(h=4\). The second derivative \(-6\pi h<0\), so this is a maximum.

Step 4: Maximum volume
\(r^{2}=48-16=32\), so \(V=\pi\cdot32\cdot4=128\pi\).

Final Answer:
The maximum volume is 128π cubic units. \[ \boxed{\text{(C) }128\pi} \]