Question

By dropping a stone in a quiet lake, a wave in the form of circle is generated. The radius of the circular wave increases at the rate of $2.1 \text{ cm/sec}$. Then the rate of increase of the enclosed circular region, when the radius of the circular wave is $10 \text{ cm}$ , is (Given $\pi = \frac{22}{7}$ )

• A. $66 \text{ cm}^2 / \text{ second}$
• B. $122 \text{ cm}^2 / \text{ second}$
• C. $132 \text{ cm}^2 / \text{ second}$
• D. $110 \text{ cm}^2 / \text{ second}$

Hint:

Use $\frac{dA}{dt} = 2\pi r \frac{dr}{dt}$ directly.

Answer 1

The Correct Option is C

Concept:
Area of circle: \[ A = \pi r^2 \quad \Rightarrow \quad \frac{dA}{dt} = 2\pi r \frac{dr}{dt} \] Step 1: Substitute values. \[ r = 10 \text{ cm}, \quad \frac{dr}{dt} = 2.1 \] \[ \frac{dA}{dt} = 2 \cdot \frac{22}{7} \cdot 10 \cdot 2.1 \]
Step 2: Calculate. \[ = \frac{44}{7} \cdot 21 = 44 \cdot 3 = 132 \]
Step 3: Conclusion. \[ {132 \text{ cm}^2/\text{s}} \]