Chapter 12.8, Problem 29STP

To determine

To find: the prism which is not similar to the other three prisms.

Answer to Problem 29STP

The prism $B$ is not similar to other three prisms.

Explanation of Solution

Given:

Prism	Length	Width	Height
$A$	$7.5$	$6$	$5$
$B$	$10.5$	$9.8$	$7$
$C$	$15$	$12$	$10$
$D$	$4.5$	$3.6$	$3$

  $F$ $A$ $H$ $C$

  $G$ $B$ $J$ $D$

Calculation:

The objective is to find the prism which is not similar to other three prisms.

Two solid figures are similar if they have the same shape and their corresponding linear measurements are proportional.

The prisms in the table are rectangular prism, so they are of same shape.

Now, check the corresponding linear measurements of prisms are proportional or not.

The table below shows the ratio of the corresponding linear measurement.

Take the prism $A$ and $B$ .

Suppose they are proportional.

  $\begin{array}{l}\frac{7.5}{10.5}=\frac{6}{9.8}\\ (7.5)(9.8)=(10.5)6\left[Crossmultiply\right]\\ 73.5=63False\end{array}$

Since the result is false, so the corresponding linear measurements of prism $A$ and $B$ are not proportional.

Next take the prism $A$ and $C$ .

Suppose the length and widths are proportional

  $\begin{array}{l}\frac{7.5}{15}=\frac{6}{12}\\ \left(7.5\right)\left(12\right)=\left(6\right)15\\ 90=90\end{array}$

Since the result is true, so the corresponding linear measurements of prism $A$ and $C$ are proportional.

Again, the length and height are proportional

  $\begin{array}{l}\frac{7.5}{15}=\frac{5}{10}\\ \left(7.5\right)\left(10\right)=\left(5\right)15\\ 75=75\end{array}$

Since the result is true, so the corresponding linear measurements of prism $A$ and $C$ are proportional.

Next take the prism $A$ and $D$ .

Suppose the length and widths are proportional.

  $\begin{array}{l}\frac{7.5}{4.5}=\frac{6}{3.6}\\ \left(7.5\right)\left(3.6\right)=\left(4.5\right)6\\ 27=27\end{array}$

Since the result is true, so the corresponding linear measurements of prism $A$ and $D$ are proportional.

Again, the length and height are proportional

  $\begin{array}{l}\frac{7.5}{4.5}=\frac{5}{3}\\ \left(7.5\right)\left(3\right)=\left(4.5\right)5\\ 22.5=22.5\end{array}$

Since the result is true, so the corresponding linear measurements of prism $A$ and $D$ are proportional.

Therefore, the prism $B$ is not similar to other three prisms. Therefore, the correct choice is $(G)$ .

Conclusion:

Therefore, the prism $B$ is not similar to other three prisms.