Chapter 1.7, Problem 19E

Solution

To write: The specified quantity as a function of specified variable.

The required function is $S=24r^2$

Given data: A sphere is contained in a cube, tangent to all six faces. The surface area (the specified quantity) is a function of the radius (specified variable) of the sphere.

Method/Formula used:

The surface area of a cube of edge ‘ a’ is $6a^2$ .

Calculation:

The sphere of radius ‘ r’ is contained in a cube of edge ‘ a’ as shown in Fig. (1).

  

As shown in the Fig. (1), the diameter AB of the sphere is equal to the edge of the cube. If $a$ is the edge of the cube and $r$ the radius of the sphere, then

  $\begin{array}{l}AB=a\\ 2r=a\end{array}$

Denote the surface area of the cube by S , then

  $S=6a^2$ … (1)

Substitute $2r$ for $a$ in equation (1),

  $\begin{array}{c}S=6{\left(2r\right)}^2\\ =6\cdot 4r^2\\ =24r^2\end{array}$

Thus, the surface area of the cube as a function of radius of the sphere is given by $S=24r^2$