Chapter 13.5, Problem 5PSA

(a)

To determine

To find:The area of the base.

Answer to Problem 5PSA

The area of the base is 20.

Explanation of Solution

Given:

The vertices is given as,

  $\begin{array}{l}A=\left[4,0,0\right]\\ B=\left[4,5,0\right]\\ C=\left[0,5,0\right]\\ D=\left[0,0,0\right]\end{array}$

The height is 7

Calculation:

The graph on the 3-D plane is given as,

  

Calculate the area of the base.

  $\begin{array}{c}{A}_{base}=4\left(5\right)\\ =20\end{array}$

Thus, the area of the base is 20.

(b)

To determine

To find:The volume of the solid

Answer to Problem 5PSA

The volume of the solid is 140.

Explanation of Solution

Given:

The vertices is given as,

  $\begin{array}{l}A=\left[4,0,0\right]\\ B=\left[4,5,0\right]\\ C=\left[0,5,0\right]\\ D=\left[0,0,0\right]\end{array}$

The height is 7

Calculation:

Calculate volume of the solid.

  $\begin{array}{c}{V}_{base}=4\left(5\right)\left(7\right)\\ =20\left(7\right)\\ =140\end{array}$

Thus, the volume of the solid is 140.

(c)

To determine

To find:The diagonal of the solid

Answer to Problem 5PSA

The diagonal of the solid is $3\sqrt{10}$ .

Explanation of Solution

Given:

The vertices is given as,

  $\begin{array}{l}A=\left[4,0,0\right]\\ B=\left[4,5,0\right]\\ C=\left[0,5,0\right]\\ D=\left[0,0,0\right]\end{array}$

The height is 7

Calculation:

Calculate diagonal of the solid.

  $\begin{array}{c}d=\sqrt{{\left(4-0\right)}^2+{\left(5-0\right)}^2+{\left(7-0\right)}^2}\\ =\sqrt{16+25+49}\\ =\sqrt{90}\\ =3\sqrt{10}\end{array}$

Thus, the diagonal of the solid is $3\sqrt{10}$ .

(d)

To determine

To check:Whether the (4,5,7) is the vertex of solid.

Answer to Problem 5PSA

Yes, the points (4,5,7) is the vertex of solid.

Explanation of Solution

Given:

The vertices is given as,

  $\begin{array}{l}A=\left[4,0,0\right]\\ B=\left[4,5,0\right]\\ C=\left[0,5,0\right]\\ D=\left[0,0,0\right]\end{array}$

The height is 7

Calculation:

Consider the figure in part (a), and it is concluded that yes, (4,5,7) is the vertex of the solid.

Thus, Yes, the points (4,5,7) is the vertex of solid.

(e)

To determine

To find: The new coordinates of the vertex.

Answer to Problem 5PSA

The new coordinates of the vertex is (11,5,0) or (-3,5,0)

Explanation of Solution

Given:

The vertices is given as,

  $\begin{array}{l}A=\left[4,0,0\right]\\ B=\left[4,5,0\right]\\ C=\left[0,5,0\right]\\ D=\left[0,0,0\right]\end{array}$

The height is 7

Calculation:

When the solid is rotating by 90, then the x coordinates is shifted by +-7 and z coordinates becomes zero, then it becomes (11,5,0) or (-3,5,0)

Thus, the new coordinates of the vertex is (11,5,0) or (-3,5,0)