Chapter 11.5, Problem 25HP

To determine

To Prove: The find the area of the circle with the radius of $r$ . Also, determine if radius is enlarged by the factor $k$ then the form is correct or not.

Answer to Problem 25HP

The form when the radius is enlarged by the factor $k$ is incorrect and the area of the circle is $\pi k^2{r}^2$ .

Explanation of Solution

Given:

The circle is with radius $r$

Calculation:

Consider the case when the radius is multiplied by the factor $k$ then the radius is $kr$ and the area is,

  $\begin{array}{c}A=\pi {\left(kr\right)}^2\\ =\pi k^2{r}^2\end{array}$

Since, $\pi k^2{r}^2\ne k\pi r^2$ and $k^2\pi r^2\ne \pi {\left(r^2\right)}^k$ .

Thus, the form when the radius is enlarged by the factor $k$ is incorrect and the area of the circle is $\pi k^2{r}^2$ .