Question

The radius of a circle is increasing at a nonzero rate, and at a certain instant, the rate of increase in the area of the circle is numerically equal to the rate of increase in its circumference. At this instant, the radius of the circle is _________.

Answer 1

Answer:

r=1

So,radius of the circle,when the rate of increase in the area of the circle is numerically equal to the rate of increase in its circumference, is 1.

Step-by-step explanation:

Area of the circle=$A=\pi r^{2}$

Circumference of circle=$C=2\pi r$

Both area and circumference of circle is increasing at the same rate.

Increase in Area of the circle=Increase in Circumference of the circle

$\frac{dA}{dt} =\frac{dC}{dt}$

$\frac{d(\pi*r^{2})}{dt} =\frac{d(2*\pi*r)}{dt}$

$2*\pi *r*\frac{dr}{dt}= 2*\pi *\frac{dr}{dt}$

By cancelling the common terms we will get:

r=1

So,radius of the circle,when the rate of increase in the area of the circle is numerically equal to the rate of increase in its circumference, is 1.

Answer 2

The radius of the circle will be 1 unit when the rate of increase in the area of the circle is numerically equal to the rate of increase in its circumference.

Given information:

The radius of a circle is increasing at a nonzero rate, and at a certain instant, the rate of increase in the area of the circle is numerically equal to the rate of increase in its circumference.

Let the radius of the circle be r.

The rate of change of circumference will be,

$\dfrac{d(2\pi r)}{dt}=2\pi\dfrac{dr}{dt}$

The rate of change of area of the circle will be,

$\dfrac{d(\pi r^2)}{dt}=\dfrac{d(\pi r^2)}{dr}\times \dfrac{dr}{dt}\\=2\pi r \dfrac{dr}{dt}$

Now, if the rate of change of area is equal to the rate of change of circumference, the value of radius can be calculated as,

$2\pi r \dfrac{dr}{dt}=2\pi \dfrac{dr}{dt}\\r=1$

Therefore, the radius of the circle will be 1 unit when the rate of increase in the area of the circle is numerically equal to the rate of increase in its circumference.

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https://brainly.com/question/19251155