Chapter 12.2, Problem 27IP

(a)

To determine

Tocomplete:Thetable.

Answer to Problem 27IP

Thecomplete table is given below.

Explanation of Solution

Given information:

Consider the given table.

Radius $\left(\text{cm}\right)$	Area $\left({\text{cm}}^2\right)$
3
6
12
24
48

Calculation:

Write the expression to calculate the area of the circle.

  $A=\pi r^2$

Where,

• $r$ is the radius of the circle.

Use the above formula to complete the table.

Radius $\left(\text{cm}\right)$	Area $\left({\text{cm}}^2\right)$
3	$\pi {\left(3\right)}^2=9\pi$
6	$\pi {\left(6\right)}^2=36\pi$
12	$\pi {\left(12\right)}^2=144\pi$
24	$\pi {\left(24\right)}^2=576\pi$
48	$\pi {\left(48\right)}^2=2304\pi$

Conclusion:

Therefore, the complete table is shown above.

(b)

To determine

Tofind:Thechange in the area of the circle when radius becomes double.

Answer to Problem 27IP

Thearea of the circle changes 4 times when radius becomes double.

Explanation of Solution

Given information:

From part (a), the radius and area is given below.

Radius $\left(\text{cm}\right)$	Area $\left({\text{cm}}^2\right)$
3	$\pi {\left(3\right)}^2=9\pi$
6	$\pi {\left(6\right)}^2=36\pi$
12	$\pi {\left(12\right)}^2=144\pi$
24	$\pi {\left(24\right)}^2=576\pi$
48	$\pi {\left(48\right)}^2=2304\pi$

Calculation:

From the table, it is clear that the area of the circle changes 4 times when radius becomes double.

Conclusion:

Therefore, the area of the circle changes 4 times when radius becomes double.

(c)

To determine

Topredict:Thearea of the circle when the radius of the circle is 96 centimeters. Verify it.

Answer to Problem 27IP

Thearea of the circleis $9216\pi {\text{cm}}^2$ .

Explanation of Solution

Given information:

From part (a), the radius and area is given below.

Radius $\left(\text{cm}\right)$	Area $\left({\text{cm}}^2\right)$
3	$\pi {\left(3\right)}^2=9\pi$
6	$\pi {\left(6\right)}^2=36\pi$
12	$\pi {\left(12\right)}^2=144\pi$
24	$\pi {\left(24\right)}^2=576\pi$
48	$\pi {\left(48\right)}^2=2304\pi$

Calculation:

From the table, it is clear that the area of the circle changes 4 times when radius becomes double. Since, the area of the circle is $2304\pi {\text{cm}}^2$ when the radius is 48 centimeters. When the radius becomes 96 i.e. double, the area of the circle will be 4 times i.e. $9216\pi {\text{cm}}^2$ .

To verify the area, calculate the area of the circle using the formula.

  $\begin{array}{c}\text{Area}=\pi {\left(96\right)}^2\\ =9216\pi {\text{cm}}^2\end{array}$

From the above, it is clear that the area of the circle is similar in both the cases.

Conclusion:

Therefore, the area of the circle is $9216\pi {\text{cm}}^2$ .