Question

adair advertising has 2 spherical balloons. one has a radius of 3 feet and the other one has a radius of 5 feet. what is the difference in the volume of the two balloons, rounded to the nearest tenth of a cubic foot? use 3.14 for π. 410.3 ft³ 201.1 ft³ 67.0 ft³ 205.3 ft³

Explanation:

Step1: Recall the volume formula for a sphere

The volume \( V \) of a sphere is given by the formula \( V = \frac{4}{3}\pi r^3 \), where \( r \) is the radius of the sphere.

Step2: Calculate the volume of the balloon with radius 5 feet

Substitute \( r = 5 \) and \( \pi = 3.14 \) into the formula:
\[

$$\begin{align*} V_1&=\frac{4}{3}\times3.14\times5^3\\ &=\frac{4}{3}\times3.14\times125\\ &=\frac{4\times3.14\times125}{3}\\ &=\frac{1570}{3}\\ &\approx523.333 \end{align*}$$

\]

Step3: Calculate the volume of the balloon with radius 3 feet

Substitute \( r = 3 \) and \( \pi = 3.14 \) into the formula:
\[

$$\begin{align*} V_2&=\frac{4}{3}\times3.14\times3^3\\ &=\frac{4}{3}\times3.14\times27\\ &=4\times3.14\times9\\ & = 113.04 \end{align*}$$

\]

Step4: Find the difference in volumes

Subtract the smaller volume from the larger volume:
\[

$$\begin{align*} \text{Difference}&=V_1 - V_2\\ &\approx523.333- 113.04\\ &\approx410.3 \end{align*}$$

\]

Answer:

\( 410.3\space\text{ft}^3 \) (corresponding to the option "410.3 \( \text{ft}^3 \)")