Chapter 12.6, Problem 46HP

To determine

Find that the statement is true or false.

Answer to Problem 46HP

The statement is true.

Explanation of Solution

Given:

It is given in the question that the statement is “If a sphere has radius r, there exists a cone with radius r having the same volume”.

Here, the statement is true. The volume of the cone $V_{cone}=\frac{1}{3}\pi r^{2}h$ and the volume of a sphere is $V_{sphere}=\frac{4}{3}\pi r^3$ . It has to be see that the volume and the radius to be the same. So, set the formulas equal to each other and solve for h .

  $\begin{array}{l}\frac{4}{3}\pi r^3=\frac{1}{3}\pi r^{2}h\\ \frac{4}{3}{r}^3=\frac{1}{3}{r}^{2}h(removing\pi )\\ \frac{4}{3}r=\frac{1}{3}h(removingr)\\ 4r=h(multiplying3)\\ h=4r\end{array}$

It shows that for a given radius of the sphere it has to pick the height of the cone to be $4r$ .