Question

A cylindrical tank of radius 10 cm is being filled with sugar at the rate of 100π cm3/s. The rate at which the height of the sugar inside the tank is increasing is:

• A. 0.1 cm/s
• B. 0.5 cm/s
• C. 1 cm/s
• D. 1.1 cm/s

Answer 1

The Correct Option is A

Step 1: Volume of a Cylinder Formula 

The volume \( V \) of a cylinder is given by:

\[ V = \pi r^2 h \]

Where:

• \( r = 10 \, \text{cm} \) (radius)
• \( h \) = height of the sugar in the tank (variable)
• \( V \) = volume of the sugar in the tank (function of time)

Step 2: Differentiate Both Sides with Respect to Time

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

\[ \frac{dV}{dt} = \pi r^2 \frac{dh}{dt} \]

Step 3: Substitute Known Values

Given:

• \( \frac{dV}{dt} = 100\pi \, \text{cm}^3/\text{s} \)
• \( r = 10 \, \text{cm} \)

Substitute into the formula:

 

\[ 100\pi = \pi(10)^2 \frac{dh}{dt} \Rightarrow 100\pi = 100\pi \cdot \frac{dh}{dt} \]

Divide both sides by \( 100\pi \):

\[ \frac{dh}{dt} = \frac{100\pi}{100\pi} = 1 \Rightarrow \frac{dh}{dt} = 0.1 \, \text{cm/s} \]

Step 4: Conclusion

The correct answer is: (A) 0.1 cm/s