Question

An ice cream store has the pricing shown below. You want to determine the best value. The height of the cone is 4.5 in and the diameter is 2 in. The diameter of each scoop is 3 in. Assume the cone is stuffed full with ice cream.Question: Create a table showing the total cost of each size, the total volume of ice cream for each size, and the unit cost (per unit volume) of each size. Include units with your table entries.

Answer 1

The cone has a height of h = 4.5 in and a diameter of d = 2 in.

The radius is r = d/2 = 1 in.

The volume of a cone is:

`V_c=(\pi r^2h)/(3)`

Substituting:

`\begin{gathered} V_c=(\pi(1)^2(4.5))/(3) \\ V_c=1.5\pi\text{ }in^3 \end{gathered}`

We assume each scoop of ice cream is a semisphere of diameter d = 3 in, or radius r = 1.5 in.

The volume of each scoop is:

`\begin{gathered} V_s=(2)/(3)\pi r^3 \\ \\ V_s=(2)/(3)\pi1.5^3=2.25\pi\text{ }in^3 \end{gathered}`

The Single size has one scoop of ice cream, thus the total volume is:

1.5π + 2.25π = 3.75π cubic inches = 11.781 cubic inches.

This size costs $3.50. The unit cost is:

`(3.50)/(11.781)=0.297`

It costs approximately $0.30 per cubic inch of ice cream.

The Double size has two scoops of ice cream, thus the total volume is:

1.5π + 2 x 2.25π = 6π cubic inches = 18.850 cubic inches.

This size costs $4.50. The unit cost is:

`(4.50)/(18.850)=0.239`

It costs approximately $0.24 per cubic inch of ice cream.

The Triple size has three scoops of ice cream, thus the total volume is:

1.5π + 3 x 2.25π = 8.25π cubic inches = 25.918 cubic inches.

This size costs $5.50. The unit cost is:

`(5.50)/(25.918)=0.212`

It costs approximately $0.212 per cubic inch of ice cream.

This last size has the least price per volume of ice cream, so it's better for the customers.

The table is shown below.