Chapter 7.3, Problem 57E

(a)

To determine

The volume of the plumb bob.

Answer to Problem 57E

The volume of the plumb bob is $\frac{36\prod }{5}{\text{ cm}}^3$ .

Explanation of Solution

Given:

The radius of plumb bob is:

  $y=\frac{x}{12}\sqrt{36-x^2}$ .

Calculation:

The volume of plumb bob can be find out as:

  $\begin{array}{l}V=\prod {\displaystyle \underset{0}{\overset{6}{\int }}{\left[R(x)\right]}^2}dx\\ V=\prod {\displaystyle \underset{0}{\overset{6}{\int }}{\left[\frac{x}{12}\sqrt{36-x^2}\right]}^2}dx\\ V=\prod {\displaystyle \underset{0}{\overset{6}{\int }}\frac{x^2}{144}(36-x^2)dx}\\ V=\frac{\prod }{144}{\displaystyle \underset{0}{\overset{6}{\int }}(36x^2-x^4)dx}\\ V=\frac{\prod }{144}{\left[12x^3-\frac{x^5}{5}\right]}_0^6\\ \end{array}$

Now, substitute the values in the above equation to find the volume.

  $\begin{array}{l}V=\frac{\prod }{144}\left[12{(6)}^3-\frac{{(6)}^5}{5}\right]\\ V=\frac{\prod .{(6)}^3}{144}\left[12-\frac{36}{5}\right]\\ V=\frac{36\prod }{5}\end{array}$

Conclusion:

The volume of the plumb bob is $\frac{36\prod }{5}{\text{ cm}}^3$

(b)

To determine

The weight of the plumb bob.

Answer to Problem 57E

The weight of the plumb bob is $192g$ .

Explanation of Solution

Given:

The weight of the brass is $8.5\frac{g}{c{m}^3}$

Calculation:

  $\begin{array}{l}W=8.5\frac{g}{c{m}^3}\times V\\ W=8.5\frac{g}{c{m}^3}\times \frac{36\prod }{5}c{m}^3\\ W=192g\end{array}$

Conclusion:

The weight of the plumb bob is $192g$ .