the radius of a circle is increasing. at a certain instant, the rate of increase in the area of the circle…

the radius of a circle is increasing. at a certain instant, the rate of increase in the area of the circle is numerically equal to twice the rate of increase in its circumference. what is the radius of the circle at that instant? (a) 1/2 (b) 1 (c) √2 (d) 2 (e) 4

the radius of a circle is increasing. at a certain instant, the rate of increase in the area of the circle is numerically equal to twice the rate of increase in its circumference. what is the radius of the circle at that instant? (a) 1/2 (b) 1 (c) √2 (d) 2 (e) 4

Answer

Answer:

D. 2

Explanation:

Step1: Recall area and circumference formulas

The area of a circle is $A = \pi r^{2}$ and the circumference is $C=2\pi r$.

Step2: Differentiate with respect to time $t$

Using the chain - rule, $\frac{dA}{dt}=2\pi r\frac{dr}{dt}$ and $\frac{dC}{dt}=2\pi\frac{dr}{dt}$.

Step3: Set up the given relationship

We are given that $\frac{dA}{dt} = 2\frac{dC}{dt}$. Substitute the expressions for $\frac{dA}{dt}$ and $\frac{dC}{dt}$: $2\pi r\frac{dr}{dt}=2\times(2\pi\frac{dr}{dt})$.

Step4: Solve for $r$

Since $\frac{dr}{dt}\neq0$ (because the radius is increasing), we can cancel out $2\pi\frac{dr}{dt}$ from both sides of the equation. We get $r = 2$.