Chapter 12.5, Problem 16WE

(a)

To determine

To complete: The statement.

Answer to Problem 16WE

The complete statement is given below.

Explanation of Solution

Given information:

The larger of two similar columns is three times as high as the smaller column.

Calculation:

Let the area of the larger column is $A_1$ and the area of the smaller column is $A_2$ . Let the height of the larger column is $h_1$ and the height of the smaller column is $h_2$ . Let the radius of the larger column is $r_1$ and the radius of the smaller column is $r_2$

Since, both the columns are similar. Therefore, the ratio of height will be equal to the ratio of their radius.

  $\frac{h_1}{h_2}=\frac{r_1}{r_2}$   ... (1)

Since the height of the larger column is three times the height of the smaller column. So, the ratio of their radius from equation (1) becomes,

  $\frac{r_1}{r_2}=\frac{3}{1}$  ... (2)

Write the expression to calculate the cross section of the larger column.

  $A_1=\pi r_1{}^2$   ... (3)

Write the expression to calculate the cross section of the smaller column.

  $A_2=\pi r_2{}^2$   ... (4)

Divide equation (3) by equation (4).

  $\begin{array}{c}\frac{A_1}{A_2}=\frac{\pi r_1{}^2}{\pi r_2{}^2}\\ ={\left(\frac{r_1}{r_2}\right)}^2\end{array}$

Substitute the value in the above expression from equation (2).

  $\begin{array}{c}\frac{A_1}{A_2}={\left(\frac{3}{1}\right)}^2\\ \frac{A_1}{A_2}=\frac{9}{1}\\ A_1=9A_2\end{array}$

Therefore, the larger column is 9 times strong as the smaller column.

Conclusion:

Therefore, the complete statement is “The larger column is 9 times strong as the smaller column.”

(b)

To determine

To complete: The statement.

Answer to Problem 16WE

The complete statement is given below.

Explanation of Solution

Given information:

The larger of two similar columns is three times as high as the smaller column.

Calculation:

Since, both the columns are similar. Therefore, the ratio of height will be equal to the ratio of their radius.

  $\frac{h_1}{h_2}=\frac{r_1}{r_2}$   ... (1)

Since the height of the larger column is three times the height of the smaller column. So, the ratio of their radius from equation (1) becomes,

  $\frac{r_1}{r_2}=\frac{3}{1}$  ... (2)

Write the expression to calculate the volume of the larger column.

  $V_1=\pi r_1{}^2{h}_1$   ... (3)

Write the expression to calculate the cross section of the smaller column.

  $V_2=\pi r_2{}^2{h}_2$   ... (4)

Divide equation (3) by equation (4).

  $\begin{array}{c}\frac{V_1}{V_2}=\frac{\pi r_1{}^2{h}_1}{\pi r_2{}^2{h}_2}\\ ={\left(\frac{r_1}{r_2}\right)}^2\left(\frac{h_1}{h_2}\right)\end{array}$

Substitute the value in the above expression from equation (2).

  $\begin{array}{c}\frac{V_1}{V_2}={\left(\frac{3}{1}\right)}^2\left(\frac{3}{1}\right)\\ \frac{V_1}{V_2}={\left(\frac{3}{1}\right)}^3\\ V_1=27V_2\end{array}$

Therefore, the larger column is 27 times as heavy as the smaller column.

Conclusion:

Therefore, the complete statement is “The larger column is 27 times as heavy as the smaller column.”

(c)

To determine

To check: Whether the larger column will support more or smaller column.

Answer to Problem 16WE

The larger column will support more.

Explanation of Solution

Given information:

The larger of two similar columns is three times as high as the smaller column.

Calculation:

From part (b), the weight of the larger column is 27 times as strong as the smaller column. Therefore, the amount of material is more in larger column. Therefore, the larger column will support more.

Conclusion:

Therefore, the larger column will support more.