Find the average rate of change of the area of a circle with respect to its radius r as r changes from to .

Question

Find the average rate of change of the area of a circle with respect to its radius r as r changes from  to  .

Quizplus · Accepted Answer

The formula for the area of a circle is $A=\pi r^2$. We can find the average rate of change of the area with respect to radius by taking the derivative of this formula with respect to $r$:

$\frac{dA}{dr}=2\pi r$

This means that for every increase in radius of 1 unit, the area of the circle increases by $2\pi$ square units.

To find the average rate of change of the area as $r$ changes from $11eaa3e5_55ee_7323_9f8f_13ea7cc8c69d\_TB5971\_11$ to $11eaa3e5_55ee_9a34_9f8f_bd5d3d860dbc\_TB5971\_11$, we need to find the difference in the area for a 1 unit increase in radius:

$A(11eaa3e5_55ee_9a34_9f8f_bd5d3d860dbc\_TB5971\_11)-A(11eaa3e5_55ee_7323_9f8f_13ea7cc8c69d\_TB5971\_11)$

$=\pi (11eaa3e5_55ee_9a34_9f8f_bd5d3d860dbc\_TB5971\_11)^2 - \pi (11eaa3e5_55ee_7323_9f8f_13ea7cc8c69d\_TB5971\_11)^2$

$= \pi(11eaa3e5_55ee_c147_9f8f_978ead83e3ef\_TB5971\_11 - 11eaa3e5_55ee_0298_9f8f_ce2d19c1e58b\_TB5971\_11)$

$= \pi(11eaa3e5_55ee_9a35_9f8f_4924f3cef37b\_TB5971\_11)$

Therefore, the average rate of change of the area with respect to radius is $\pi(11eaa3e5_55ee_9a35_9f8f_4924f3cef37b\_TB5971\_11)$ square units per unit. Simplifying this answer gives us choice D.

Find the Average Rate of Change of the Area of a Circle