Chapter 9.4, Problem 19WE

To determine

To Find : The diameter of sphere.

Answer to Problem 19WE

The measure of diameter of sphere CD is 26cm.

Explanation of Solution

Given information:

A plane 5cm from the center of a sphere intersects the sphere in a circle with a diameter 24cm.

Consider the figure as below:

  

In above figure a plane AB is 5cm from the centre point O of the sphere and it intersects in a circle with diameter AB=24.

In sphere diameter CD bisects AB=24 into two equal parts from the midpoint M which means $MA=MB$ and $AB=MA+MB$

Or it is define as

  $MB=\frac{AB}{2}$

If $AB=24$ then

  $\begin{array}{l}MB=\frac{AB}{2}\hfill \\ MB=\cancel{\frac{24}{2}}\hfill \\ MB=12\hfill \end{array}$

Let $OB=x$ this is a radius of the sphere.

Use Pythagoras theorem:

  $\left(Hypotenuse\right)²=\left(Perpendicular\right)²+\left(Base\right)²$

In triangle ΔOMB

Here Hypotenuse OB= x, Perpendicular OM= 5 and Base MB=12

  $\begin{array}{l}\left(x\right)²=\left(5\right)²+\left(12\right)²\hfill \\ \left(x\right)²=25+144\hfill \\ \begin{array}{l}\left(x\right)²=169\\ x=13\end{array}\hfill \end{array}$

Hence, radius of sphere $OB=13$ .

This means other radius of the sphere OC and OD will also equal to 13 .

Then diameter of sphere CD will be

  $\begin{array}{l}CD=OC+OD\hfill \\ CD=13+13\hfill \\ CD=26\hfill \end{array}$

Therefore, measure of diameter of sphere CD is 26 cm.