To find the difference in the volume of the two balloons, we need to follow several steps. Here are the calculations:
1. Volume Formula for a Sphere:
The volume [tex]\( V \)[/tex] of a sphere is given by the formula:
[tex]\[
V = \frac{4}{3} \pi r^3
\][/tex]
where [tex]\( r \)[/tex] is the radius of the sphere and [tex]\( \pi \)[/tex] is approximately 3.14.
2. Calculating the Volume of the First Balloon (radius = 3 feet):
[tex]\[
V_1 = \frac{4}{3} \pi (3)^3
\][/tex]
[tex]\[
V_1 = \frac{4}{3} \times 3.14 \times 27
\][/tex]
[tex]\[
V_1 \approx 113.04 \, \text{cubic feet}
\][/tex]
3. Calculating the Volume of the Second Balloon (radius = 5 feet):
[tex]\[
V_2 = \frac{4}{3} \pi (5)^3
\][/tex]
[tex]\[
V_2 = \frac{4}{3} \times 3.14 \times 125
\][/tex]
[tex]\[
V_2 \approx 523.33 \, \text{cubic feet}
\][/tex]
4. Finding the Difference in Volume Between the Two Balloons:
[tex]\[
\text{Difference} = V_2 - V_1
\][/tex]
[tex]\[
\text{Difference} = 523.33 - 113.04
\][/tex]
[tex]\[
\text{Difference} \approx 410.29 \, \text{cubic feet}
\][/tex]
5. Rounding the Difference to the Nearest Tenth:
The difference in volume rounded to the nearest tenth is:
[tex]\[
\text{Difference} \approx 410.3 \, \text{cubic feet}
\][/tex]
Therefore, the difference in the volume of the two balloons, rounded to the nearest tenth of a cubic foot, is [tex]\(\boxed{410.3 \, \text{ft}^3}\)[/tex].
