The radius of a circular oil slick expands at a rate of .
(a) How fast is the area of the oil slick increasing when the radius is 22 m ?

(b) If the radius is 0 at time , how fast is the area increasing after 3 mins ?

Ask by Ingram Powers. in the United States

Mar 15,2025

Question

The radius of a circular oil slick expands at a rate of .
(a) How fast is the area of the oil slick increasing when the radius is 22 m ?

(b) If the radius is 0 at time , how fast is the area increasing after 3 mins ?

Ask by Ingram Powers. in the United States

Mar 15,2025

UpStudy AI · Accepted Answer

We are given that the radius $$ r $$ of the circular oil slick increases at a constant rate of  
$$
\frac{dr}{dt} = 4 \, \text{m/min}.
$$
The area $$ A $$ of a circle is given by  
$$
A = \pi r^2.
$$
We differentiate with respect to time $$ t $$ using the chain rule:  
$$
\frac{dA}{dt} = \frac{d}{dt}(\pi r^2) = 2\pi r \frac{dr}{dt}.
$$

**(a)** When the radius is $$ r = 22 \, \text{m} $$:

Substitute $$ r = 22 $$ and $$ \frac{dr}{dt} = 4 $$ into the differentiated formula:  
$$
\frac{dA}{dt} = 2\pi (22)(4) = 176\pi \, \text{m}^2/\text{min}.
$$

So,  
$$
\frac{dA}{dt} = 176\pi \, \text{m}^2/\text{min} \quad \text{when} \quad r = 22 \, \text{m}.
$$

**(b)** When $$ r = 0 $$ at $$ t = 0 $$ and $$ \frac{dr}{dt} = 4 \, \text{m/min} $$:

Since the radius increases at 4 m/min, the radius after $$ t $$ minutes is  
$$
r = 4t.
$$
After 3 minutes,  
$$
r = 4(3) = 12 \, \text{m}.
$$
Substitute $$ r = 12 $$ and $$ \frac{dr}{dt} = 4 $$ into the formula:  
$$
\frac{dA}{dt} = 2\pi (12)(4) = 96\pi \, \text{m}^2/\text{min}.
$$

Thus,  
$$
\frac{dA}{dt} = 96\pi \, \text{m}^2/\text{min} \quad \text{after 3 minutes}.
$$
**(a)** When the radius is 22 meters, the area is increasing at $$ 176\pi \, \text{m}^2/\text{min} $$. **(b)** After 3 minutes, the area is increasing at $$ 96\pi \, \text{m}^2/\text{min} $$.

The radius of a circular oil slick expands at a rate of .
(a) How fast is the area of the oil slick increasing when the radius is 22 m ?

(b) If the radius is 0 at time , how fast is the area increasing after 3 mins ?

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The radius of a circular oil slick expands at a rate of . (a) How fast is the area of the oil slick increasing when the radius is 22 m ? (b) If the radius is 0 at time , how fast is the area increasing after 3 mins ?

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Latest Calculus Questions

The radius of a circular oil slick expands at a rate of .
(a) How fast is the area of the oil slick increasing when the radius is 22 m ?

(b) If the radius is 0 at time , how fast is the area increasing after 3 mins ?