Question

You are trying to figure out how many gumballs you need to fill a 4.9 x6.5x5.6 rectangular box for Halloween. Each gumball has a radius of 1/2 in, if the packing density forspheres 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.Type your answer

Answer 1

`213\text{ balls}`

Step-by-step explanation:

The dimensions of the box = 4.9 by 6.5 by 5.6

We would the dimensions to find the volume of the rectangular box:

`\begin{gathered} \text{Volume = }4.9*6.5*5.6 \\ \text{Volume of the rectangular box = 178.36} \end{gathered}`

radius of gumball = 1/2 in = 0.5 in

gumball are spherical in shape, so we find the volume of the gumball:

`\begin{gathered} \text{Volume of sphere = }(4)/(3)^{}\pi r^3 \\ \text{Volume of one gumball = }(4)/(3)^{}\pi(0.5)^3 \\ \text{let }\pi\text{= 3.14} \\ \text{Volume of one gum ball = 0.5233 in}^3 \end{gathered}`

Packing density of spheres = 5/8

This means 5/8 of the rectangular box will contain the gumballs while the rest will be air

`\begin{gathered} \text{Volume of the space occupied by gumball = 5/8 (volume of rectangular box)} \\ \text{Volume of the space occupied by gumball =}(5)/(8)(178.36) \\ \text{Volume of the space occupied by gumball =}111.475 \end{gathered}`

Number of gumballs needed:

`\begin{gathered} =\text{ }\frac{Volume\text{ of space occupied by gumball}}{\text{volume of one gumball}} \\ =\text{ }(111.475)/(0.5233) \\ =\text{ 213.02} \\ To\text{ the nearest whole number, } \\ \text{number of gumballs n}eeded\text{ = }213 \end{gathered}`