Question

You are trying to figure out how many gumballs you need to fill a 8.8in x 8.5in x 7.3in rectangular box for Halloween. Each gumball has a radius of 1/2 in, if the packing density for spheres is 5/8 of the volume will be filled with gumballs while the rest will be air how many gumballs will be needed? Round to the nearest whole number

Answer 1

Given:

The dimensions of the rectangular box are;

`8.8in*8.5in*7.3in`

The volume of the rectangular box will be;

`\begin{gathered} \text{The volume of the rectangular box is calculated by;} \\ V=\text{lbh} \\ \\ \text{Hence,} \\ V=8.8*8.5*7.3 \\ V=546.04in^3 \end{gathered}`

The gumball is assumed to be spherical;

`\begin{gathered} \text{The volume of a sphere is given by;} \\ V=(4)/(3)\pi r^3 \\ \text{where r is the radius } \\ r=(1)/(2) \\ \text{Hence,} \\ V=(4)/(3)*\pi*((1)/(2))^3 \\ V=(4)/(3)*\pi*(1)/(8) \\ V=(\pi)/(6)in^3 \end{gathered}`

The packing density for spheres is 5/8 of the volume while the rest is air;

`\begin{gathered} \text{Hence, the volume of the gumball to be in the box will be;} \\ (5)/(8)\text{ of the volume of the gumball} \\ =(5)/(8)*(\pi)/(6) \\ =(5\pi)/(48)in^3 \end{gathered}`

Hence, the number of gumballs considering 5/8 of the volume that will fill the rectangular box will be;

`\begin{gathered} \frac{\text{volume of rectangular box}}{\text{volume of gumball}} \\ =(564.04)/((5\pi)/(48)) \\ =564.04*(48)/(5\pi) \\ =1668.57533 \\ \approx1669\text{ gumballs to the nearest whole number} \end{gathered}`

Therefore, the number of gumballs needed to fill the rectangular box to the nearest whole number is 1669.