If the rate of change of area of a circle is equal to the rate of change of its diameter, then its radius is equal to (a) unit (b) unit (c) units (d) units

Options

A. `(2)/(pi)` unit
B. `(1)/(pi)` unit
C. `(pi)/(2)` units
D. `pi` units

Explanation

To solve the problem, we need to find the radius of a circle when the rate of change of its area is equal to the rate of change of its diameter. Let's go through the steps systematically.

### Step 1: Understand the formulas
The area \( A \) of a circle is given by the formula:
\[ A = \pi r^2 \]
where \( r \) is the radius of the circle.

The diameter \( d \) of the circle is related to the radius by:
\[ d = 2r \]

### Step 2: Differentiate the area with respect to time
To find the rate of change of the area with respect to time, we differentiate \( A \) with respect to \( t \):
\[ \frac{dA}{dt} = \frac{d}{dt}(\pi r^2) = 2\pi r \frac{dr}{dt} \]

### Step 3: Differentiate the diameter with respect to time
Now, we differentiate the diameter with respect to time:
\[ \frac{dd}{dt} = \frac{d}{dt}(2r) = 2\frac{dr}{dt} \]

### Step 4: Set the rates equal to each other
According to the problem, the rate of change of the area is equal to the rate of change of the diameter:
\[ \frac{dA}{dt} = \frac{dd}{dt} \]
Substituting the expressions we derived:
\[ 2\pi r \frac{dr}{dt} = 2\frac{dr}{dt} \]

### Step 5: Simplify the equation
Assuming \( \frac{dr}{dt} \neq 0 \) (since if it were zero, the rates would not be changing), we can divide both sides by \( 2\frac{dr}{dt} \):
\[ \pi r = 1 \]

### Step 6: Solve for the radius
Now, we can solve for \( r \):
\[ r = \frac{1}{\pi} \]

### Conclusion
The radius of the circle when the rate of change of area is equal to the rate of change of its diameter is:
\[ r = \frac{1}{\pi} \text{ units} \]

### Final Answer
Thus, the correct option is:
(c) \( \frac{1}{\pi} \text{ units} \)

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