Question

Water is being poured at the rate of 36 m$^3$/min into a cylindrical vessel whose circular base is of radius 3 meters. Then the water level in the cylinder increases at the rate of:

• A. \(\frac{4}{\pi} \, \text{m/min} \)
• B. \(4\pi \, \text{m/min} \)
• C. \(\frac{1}{4\pi} \, \text{m/min} \)
• D. \(\frac{\pi}{4} \, \text{m/min} \)

Hint:

For problems involving cylindrical shapes, remember to use the volume formula \(V = \pi r^2 h\) and differentiate with respect to time to find the rate of change of height.

Answer 1

The Correct Option is A

The volume of a cylinder is:

\( V = \pi r^2 h \) 

Differentiate both sides with respect to time \( t \):

\( \frac{dV}{dt} = \pi r^2 \frac{dh}{dt} \)

Given:

\( \frac{dV}{dt} = 36 \, \text{m}^3/\text{min}, \quad r = 3 \, \text{m} \)

Substitute into the equation:

\( 36 = \pi (3)^2 \frac{dh}{dt} = 9\pi \frac{dh}{dt} \)

Solve for \( \frac{dh}{dt} \):

\( \frac{dh}{dt} = \frac{36}{9\pi} = \frac{4}{\pi} \)

Final Answer:

\( \boxed{\frac{4}{\pi} \, \text{m/min}} \)