Answered: "What is the rate of change of the..."

Josh Sizemore

Answered question

2021-12-24

What is the rate of change of the area of a circle with respect to the radius when the radius is r = 3 in.?

Virginia Palmer

Beginner2021-12-25Added 27 answers

The area of a circle is given as

A (r) = π r^{2}

To find the instantneous rate of change with respect to r, we apply the slope formula:

lim_{h \to 0} \frac{π {(r + h)}^{2} - π r^{2}}{h} = lim_{h \to 0} \frac{π (r^{2} + 2 h r + h^{2}) - π r^{2}}{h}

= lim_{h \to 0} \frac{π (2 h r + h^{2})}{h}

= lim_{h \to 0} \frac{π h (2 r + h)}{h}

= lim_{h \to 0} [π (2 r + h)]

Substitute

r = 3

= lim_{h \to 0} [π (6 + h)]

= 6 π

When the radius is 3 in., area is changing at a rate of

6 π

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