Question

the radius of a circle is increasing at the rate of 5.5 cm per second at what rate is the area increasing when the radius of the circle is 6 CM​

Answer 1

Answer:

The radius of a circle is increasing at the rate of 5.5 cm per second at what rate is the area increasing when the radius of the circle is 6 CM​.

The area of the circle is increasing at the rate of 397.463 cm²/second

Step-by-step explanation:

If the radius of the circle increases at the rate of 5.5 cm per second, then let us take the circle at two instances.

If the radius is 6 cm and the increase at the rate of 5.5cm/s, let us calculate the change in radius after 2 seconds

If 1 second  = 5.5 cm

Then 2 seconds  = 5.5 cm × 2 s/1s

                             11 cm

Therefore, the radius increase by 11 cm after 2 seconds

The new radius after 2 seconds = 6 cm +  11 cm = 17 cm

Find the area with the two radii and calculate the rate of change:

Area of circle = πr²

Area₁  = 3.142 × 6cm² = 113.112 cm²

Area₂ = 3.142 × 17cm² = 908.038 cm²

Rate of change in the area = change in area/ change in time

                                             = (908.038cm²  - 113.112cm²) / 2 seconds

                                             = 794.926 cm²/ 2 seconds

                                             = 397.463 cm²/second

Therefore the rate of increase in area of the circle is 397.463 cm²/second

Answer 2

Answer:

The area is increasing at the of 207.24 square cm/sec

Step-by-step explanation:

Let r and A be the radius and area of the circle at time t.

Given:

$\frac{dr}{dt}=5.5\:cm/sec$

To find:

$\frac{dA}{dt}$ when r=6 cm

Now,

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

Differentiate with respect 't'

$\frac{dA}{dt}=\pi\:2r\:\frac{dr}{dt}$

$\frac{dA}{dt}=\pi\:2r\:\frac{dr}{dt}$

$\frac{dA}{dt}=\pi\:2(6)(5.5)$

$\frac{dA}{dt}=66\:\pi$

$\frac{dA}{dt}=66(3.14)$

$\implies\:\frac{dA}{dt}=207.24\:cm^2/sec$