Chapter 15, Problem 47CR

a.

To determine

To explain: the locus in space of points generated by obtuse $\Delta ABC$ .

Answer to Problem 47CR

The locus is a cone with a radius of 15, with a conical hole of radius 6 inside it.

Explanation of Solution

Given:

The given obtuse triangle $\Delta ABC$ is rotated about the altitude from A to BC is shown below figure.

  

Calculation:

According to the question, the locus in space of points generated by obtuse $\Delta ABC$ .

Thus, the locus is a cone with a radius of 15, with a conical hole of radius 6 inside it.

b.

To determine

To find: the volume of the locus.

Answer to Problem 47CR

  $V_{locus}=504\pi$ .

Explanation of Solution

Given:

The given figure formed by joining consecutive mid-points of the sides of ABCD is shown below:

  

Calculation:

From part a, locus is a cone with a radius of 15, with a conical hole of radius 6 inside it.

Now,

  $\begin{array}{l}{V}_{cone}=\frac{1}{3}\pi A_{bsae}\times h\\ V_{locus}=V_{\text{large}\text{cone}}-V_{\text{small}\text{cone}}\\ \therefore V_{locus}=\frac{1}{3}\pi r_l^2\times h-\frac{1}{3}\pi A{r}_s^2\times h\\ V_{locus}=\frac{1}{3}\pi \times h\left(r_l^2-r_s^2\right)\end{array}$

  $\begin{array}{l}{V}_{locus}=\frac{1}{3}\pi \times 8\left({15}^2-6^2\right)\\ V_{locus}=\frac{1}{3}\pi \times 8\left(225-36\right)\\ V_{locus}=\frac{1}{3}\pi \times 8\times \left(189\right)\\ V_{locus}=8\pi \times 63=504\pi \end{array}$

Thus, the volume of the locus is $V_{locus}=504\pi$ .