Chapter 12.8, Problem 24PPS

To determine

To find the radius and height of both the cones.

Answer to Problem 24PPS

Smaller cone: $r=7$ cm, $h=21$ cm; larger cone: $r=8$ cm, $h=24$ cm

Explanation of Solution

Given:

Two similar cones having volumes $343\pi$ cubic centimeters and $512\pi$ cubic centimeters

The height of each cone is $3$ times its radius.

Concept used:

If the two solids are similar, then the ratio of the volumes of the larger and smaller solids equals the cube of the ratio of the diameters of their circular bases.

Formula used:

Volume of a cone $=\frac{1}{3}\pi r^{2}h$

Volume of a cylinder $=\pi r^{2}h$

Calculation:

Let $r$ represent the radius of the smaller cone and $3r$ represent its height.

Using the volume of the cone to find the value of $r$ .

  $\Rightarrow V_{\text{cone}}=\frac{1}{3}\pi r^{2}h$

  $\Rightarrow 343\pi =\frac{1}{3}\pi r^2\left(3r\right)$

  $\Rightarrow 343\pi =\pi r^3$

  $\Rightarrow 343=r^3$

[Dividing both sides by $\pi$ ]

  $\begin{array}{l}\Rightarrow \sqrt[3]{343}=r\\ \Rightarrow 7=r\end{array}$

So, the radius of the smaller cone is $7$ cm and its height is $3\times 7$ or $21$ cm.

Since the two cones are similar, the ratio of the volumes of the larger and smaller cones equals the cube of the ratio of the radii of the two cones.

  $\Rightarrow \frac{V_{\text{small cone}}}{V_{\text{large cone}}}={\left(\frac{r_{\text{small cone}}}{r_{\text{large cone}}}\right)}^3$

  $\Rightarrow \frac{343\pi }{512\pi }={\left(\frac{7}{r_{\text{large cone}}}\right)}^3$

  $\Rightarrow \frac{343}{512}=\frac{343}{{\left(r_{\text{large cone}}\right)}^3}$

  $\Rightarrow 512={\left(r_{\text{large cone}}\right)}^3$

  $\Rightarrow 8=r_{\text{large cone}}$

Therefore, the radius of the larger cone is $8$ cm and its height is $3\times 8$ or $24$ cm.

Conclusion:

Therefore, smaller cone: $r=7$ cm, $h=21$ cm; larger cone: $r=8$ cm, $h=24$ cm.