Chapter 12.5, Problem 4ST2

To determine

To find:The area of the circle formed by the plane.

Answer to Problem 4ST2

The area of the circleformed by the plane is $78.57{\text{cm}}^{\text{2}}$ .

Explanation of Solution

Given information:

The radius of the circle is 13 cm and distance from the center to the plane is 12 cm.

Calculation:

Consider the figure.

  

Let, the radius of the plane be $x\text{cm}$ .

The formula for Pythagoras theorem is:

  $H^2=P^2+B^2$

Substitute $13\text{cm}$ for $H$ , $12\text{cm}$ for $P$ and $x\text{cm}$ for $B$ in the above expression.

  $\begin{array}{c}{13}^2={12}^2+x^2\\ x^2={13}^2-{12}^2\\ =\left(13\times 13\right)-\left(12\times 12\right)\\ =169-144\\ =25\text{cm}\end{array}$

To find the value of radius of plane, apply square root at both sides in the above expression.

  $\begin{array}{c}\sqrt{x^2}=\sqrt{25\text{cm}}\\ x=5\text{cm}\end{array}$

Formula to calculate the area of the circle formed by plane is:

  $\text{Area}=\pi x^2$

Substitute $5$ for $x\text{cm}$ and $\frac{22}{7}$ for $\pi$ in the above expression.

  $\begin{array}{c}\text{Area}=\frac{22}{7}\times {\left(\text{5}\text{cm}\right)}^2\\ =\frac{22}{7}\times \text{5}\text{cm}\times \text{5}\text{cm}\\ =78.57{\text{cm}}^{\text{2}}\end{array}$

Therefore, the area of the circle formed by the plane is $78.57{\text{cm}}^{\text{2}}$ .