Chapter 12.4, Problem 16WE

To determine

To find: Is the come big enough to hold all the ice-cream if it melts.

Answer to Problem 16WE

The cone is not big enough to hold the ice cream if it melts.

Explanation of Solution

Given information:

The diameter of the ice-cream scoop is 6 cm. and the diameter of cone is 5 cm and height of the cone is 12 cm.

Calculation:

To find if cone is big enough to hold the ice cream if it melts compare their volume.

The Volume of ice cream scoop sphere is given by

  $V_s=\frac{4}{3}\pi r_s{}^3$ where $r_s$ is the radius of the ice cream scoop and

The Volume of ice cream cone is given by

  $V_c=\frac{1}{3}\pi r_c{}^{2}h$ where $r_c$ and $h$ is the radius and height of the cone.

Since, diameter of the ice cream scoop is 6. Therefore, its radius will be

  $\begin{array}{l}{r}_s=\frac{6}{2}\\ r_s=3\end{array}$

Calculate the volume of the ice cream scoop:

  $\begin{array}{l}{V}_s=\frac{4}{3}\pi {\left(3\right)}^3\\ V_s=\frac{4}{3}\pi \times 27\\ V_s=36\pi \end{array}$

Also, diameter of the ice cream cone is 5 and its height is 12cm. Therefore, its radius will be

  $r_c=\frac{5}{2}$

Calculate the volume of the ice cream cone:

  $\begin{array}{l}{V}_c=\frac{1}{3}\pi {\left(\frac{5}{2}\right)}^2\left(12\right)\\ V_c=\frac{1}{3}\pi \times \frac{25}{4}\times 12\\ V_c=25\pi \end{array}$

Since, volume of the ice cream scoop is greater than the volume of the ice cream cone, that is $V_s>V_c$ . Therefore, the cone is not big enough to hold the ice cream if it melts.

Hence,

The cone is not big enough to hold the ice cream if it melts.