To find the rate of change of the area of a circle when its circumference is increasing, we can use the relationships between circumference, radius, and area.
Step 1: Understand the relationship
The circumference C of a circle is given by the formula:
C = 2πr
where r is the radius. The area A of a circle is given by:
A = πr²
Step 2: Determine what’s known
We know that the circumference is increasing at a rate of 5 meters per minute:
\frac{dC}{dt} = 5 ext{ m/min}
At the moment we want to analyze, the radius is:
r = 4 ext{ m}
Step 3: Differentiate the circumference
To find the rate of change of the radius \frac{dr}{dt}, we first differentiate the circumference formula with respect to time:
\frac{dC}{dt} = 2π\frac{dr}{dt}
Now, we can solve for \frac{dr}{dt}:
\frac{dr}{dt} = \frac{1}{2π} \frac{dC}{dt}
Substituting the known rate:
\frac{dr}{dt} = \frac{1}{2π} \times 5 = \frac{5}{2π} ext{ m/min}
Step 4: Differentiate the area
Next, we differentiate the area formula:
\frac{dA}{dt} = 2πr \frac{dr}{dt}
Step 5: Substitute known values
Now we substitute r = 4 meters and \frac{dr}{dt} = \frac{5}{2π}:
\frac{dA}{dt} = 2π(4) \times \frac{5}{2π}
Calculating this gives:
\frac{dA}{dt} = 8 \times 5 = 40 ext{ m²/min}
Conclusion
Therefore, the rate of change of the area of the circle when the radius is 4 meters is:
40 square meters per minute.