Question

The radius of a circle is increasing at the rate of \( 2 \,\text{cm/sec} \). Find the rate at which its area is increasing when the radius of the circle is \( 5 \) decimeters.

• A. \( 100\pi \,\text{cm}^2/\text{sec} \)
• B. \( 200\pi \,\text{cm}^2/\text{sec} \)
• C. \( 2000\pi \,\text{cm}^2/\text{sec} \)
• D. \( 20\pi \,\text{cm}^2/\text{sec} \)

Hint:

Always convert all quantities to the same unit system before applying related-rates formulas.

Answer 1

The Correct Option is B

Step 1: Convert units.
Given radius \( r = 5 \) decimeters \( = 50 \) cm.

Step 2: Write the area formula.
\[ A = \pi r^2 \]

Step 3: Differentiate with respect to time.
\[ \frac{dA}{dt} = 2\pi r \frac{dr}{dt} \]

Step 4: Substitute the given values.
\[ \frac{dA}{dt} = 2\pi (50)(2) = 200\pi \]

Step 5: Conclusion.
\[ \boxed{200\pi \,\text{cm}^2/\text{sec}} \]