Question

The radius of a cylinder is increasing at \(2\,m/s\) and height is decreasing at \(3\,m/s\). When \(r=3\,m, h=5\,m\), rate of change of volume is:

• A. \(87\pi\, m^3/s\)
• B. \(33\pi\, m^3/s\)
• C. \(27\pi\, m^3/s\)
• D. \(15\pi\, m^3/s\)

Hint:

Use product rule when differentiating volume with respect to time when both radius and height change.

Answer 1

The Correct Option is B

Concept: Volume of cylinder: \(V = \pi r^2 h\).

Step 1: Differentiate with respect to time \(t\). \[ \frac{dV}{dt} = \pi \left( 2r \frac{dr}{dt} \cdot h + r^2 \frac{dh}{dt} \right) \]

Step 2: Substitute given values. \[ \frac{dr}{dt} = 2,\quad \frac{dh}{dt} = -3,\quad r = 3,\quad h = 5 \] \[ \frac{dV}{dt} = \pi \left( 2 \times 3 \times 2 \times 5 + (3)^2 \times (-3) \right) \] \[ = \pi \left( 2 \times 3 \times 2 \times 5 = 60,\quad 9 \times (-3) = -27 \right) \] \[ = \pi (60 - 27) = 33\pi \]