Question

The rate of change of the area of a circle with respect to its radius \( r \) at \( r = 6 \, \text{cm} \) is:

Hint:

The rate of change of the area of a circle with respect to its radius is given by \( \frac{dA}{dr} = 2\pi r \).

Answer 1

Step 1: Formula for the area of a circle.
The area \( A \) of a circle is given by the formula: \[ A = \pi r^2 \] Step 2: Differentiating with respect to \( r \).
To find the rate of change of the area with respect to \( r \), we differentiate \( A \) with respect to \( r \): \[ \frac{dA}{dr} = 2\pi r \] Step 3: Substituting \( r = 6 \, \text{cm} \).
Substituting \( r = 6 \) into the equation, we get: \[ \frac{dA}{dr} = 2\pi (6) = 12\pi \, \text{cm}^2/\text{cm} \] Step 4: Conclusion.
Thus, the rate of change of the area of the circle at \( r = 6 \, \text{cm} \) is \( 12\pi \, \text{cm}^2/\text{cm} \).