Chapter 11.5, Problem 24PPS

(a)

To determine

To Find: The copy and complete table below for the each factor of rectangular prism that is 2 inches by 3 riches by 5 inches.

Answer to Problem 24PPS

The required table is shown in Table 2.

Explanation of Solution

Given:

The given table is shown in Table 1

Table 1

Scale factor	Length	Width	Height	Volume	Ratio of selected volume to initial volume
1	3	2	3
2
3
4
5
10

Calculation:

To obtain the table each of the dimension should be multiplies by the scale and since the rectangular prism volume will be the product of all the dimensions.

The required table is shown in Table 2

Table 2

Scale factor	Length	Width	Height	Volume	Ratio of selected volume to initial volume
1	3	2	3	30	1
2	6	4	10	240	8
3	9	6	15	810	27
4	12	8	20	1920	64
5	15	10	25	3750	125
10	30	20	50	30000	1000

(b)

To determine

To Find: The conjecture about the relationship between the scale factor and the ratio of the scaled volume to the initial volume.

Answer to Problem 24PPS

The conjecture is the ratio of the scaled volume to the initial volume is the cube of the scale factor

Explanation of Solution

Calculation:

Consider the table shown in table 2

Then, the pattern for the ratio is,$\begin{array}{l}1=1^3\\ 8=2^3\\ 27=3^3\\ 64=4^3\end{array}$

Solve further as,$\begin{array}{l}125=5^3\\ 1000={10}^3\end{array}$

Then, the conjecture is the ratio of the scaled volume to the initial volume is the cube of the scale factor

(c)

To determine

To Find: The scatter plot for the scale factor and the ratio of the scaled volume by the use of STAT plot feature of the graphing utility and the represent the graph.

Answer to Problem 24PPS

The cubic regression is the best fit.

Explanation of Solution

Calculation:

In the scale factors into $L_1$ of the calculator and the ratio of the scale to the $L_2$ , then create the scatter plot by the use of $L_1$ values as the x values and the values as the value of y.

Then, press the stat, CALC and then check the differences in the regression curves.

Thus, the cubic regression is the best fit.

(d)

To determine

To Find: The algebraic expression for the ratio of the scaled volume to the initial volume in terms of scale factor $k$ .

Answer to Problem 24PPS

The required algebraic expression is $R=k^2$ .

Calculation:

Consider the ratio of the volumes as,$\begin{array}{c}\frac{\text{scaled}\text{volume}}{\text{initial}\text{volume}}=\frac{\left(kl\right)\left(kw\right)\left(kh\right)}{lwh}\\ =\frac{k^3\left(lwh\right)}{lwh}\\ =k^2\end{array}$

Thus, the required algebraic expression is $R=k^2$ .

Explanation of Solution

The required algebraic expression is $R=k^2$ .