Question

Water leaking onto a floor forms a circular pool. The radius of the pool increases at a rate of 8 cm/min. How fast is the area of the pool increasing when the radius is 13 cm?

Answer 1

Answer:

The area of the pool increasing at the rate of 653.12$cm^2/min$  when the radius is 13 cm

Step-by-step explanation:

Given:

radius of the pool increases at a rate of 8 cm/min

To Find:

How fast is the area of the pool increasing when the radius is 13 cm ?

Solution:

we are given with the circular pool

hence the area of the circular pool =

A =$\pi r^2$-----------------------------(1)

The area of the pool os increasing at the rate of 8 cm/min, meaning that the arae of the pool is changing with respect to time t

so differentiating eq (1)  with respect to t , we have

$\frac{d A}{d t}=\pi \cdot 2 r \cdot \frac{d r}{d t}$

we have to find  $\frac{d A}{d t}$ with $\frac{d r}{d t}$ = 8 cm/min and  r =  13cm

substituting the values

$\frac{d A}{d t}=\pi \cdot 2 (13) \cdot 8$

$\frac{d A}{d t}=\pi \cdot 26 \cdot 8$

$\frac{d A}{d t}=\pi \cdot 208$

$\frac{d A}{d t}= 208 \pi$

$\frac{d A}{d t}=653.12$