"The radius of a circle is increasing at..." solved and explained

Bradley Collier

Answered question

2023-01-06

The radius of a circle is increasing at a constant rate of

0.2

meters per second. what is the rate increase in the area of the circle at the instant when the circumference of the circle is

20 π

meters?

Shea Pace

Beginner2023-01-07Added 6 answers

Explanation:
Given,
The circle's radius is growing at a constant rate of

0.2 \frac{m}{s e c}

circumference of the circle at an instant is

20 π

meters
STEP-1:
Area of the circle

= π \times r^{2}

Now differentiating the equation with respect to time

t

\frac{d A}{d t} = 2 πr \frac{d r}{d t}

Given

\frac{d r}{d t} = 0.2 \frac{m}{s e c}

then,

\frac{d A}{d t} = 0.4 \times π \times r

STEP-2:
Circumference of the circle is C

= 2 πr

C

= 20 π

we get,

2 πr = 20 π \Rightarrow r = 10 m

STEP-3:
Now if we put the value of

r

in the above equation of

\frac{d A}{d t}

\Rightarrow \frac{d A}{d t} = 0.4 \times π \times 10 \Rightarrow \frac{d A}{d t} = 12.56 \frac{m^{2}}{\sec}

Therefore, the rate of increase in the area of the circle is

12.56 \frac{m^{2}}{s e c}

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