Question

The volume of gas a balloon can hold varies directly as the cube of its radius. Imagine a balloon with a radius of 3 inches can hold 81 cubic inches of gas. How many cubic inches of gas must be released to reduce the radius down to 2 inches?

Answer 1

Answer: 57 in³

Step-by-step explanation:

1. You know that the volume (V) of gas a balloon can hold varies directly as the cube of its radius (r). Then, you can write the following expression:

$V=kr^{3}$

Where k is the constant of proportionality.

2. You can calculate k as following:

$k=\frac{V_1}{r^{3}}\\k=\frac{81in^{3}}{(3in)^{3}}\\k=3$

3. Then, to calculate the cubic inches that must be released to reduce the radius down to 2 inches:

- Solve for the volume as following:

$V_2=3(2in)^{3}=24in^{3}$

- Make the subtraction of the two volumes shown above. Then, the result is:

$V_3=81in^{3}-24in^{3}=57in^{3}$