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. 205.3 ft³ 201.1 ft³ 67.0 ft³ 410.3 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 first balloon (radius \( r_1 = 3 \) feet)

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

$$\begin{align*} V_1&=\frac{4}{3}\times3.14\times3^{3}\\ &=\frac{4}{3}\times3.14\times27\\ & = 4\times3.14\times9\\ &=113.04 \end{align*}$$

\]

Step3: Calculate the volume of the second balloon (radius \( r_2 = 5 \) feet)

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

$$\begin{align*} V_2&=\frac{4}{3}\times3.14\times5^{3}\\ &=\frac{4}{3}\times3.14\times125\\ &=\frac{4\times3.14\times125}{3}\\ &=\frac{1570}{3}\\ &\approx523.33 \end{align*}$$

\]

Step4: Find the difference in volumes

Subtract the volume of the smaller balloon from the volume of the larger balloon:
\[

$$\begin{align*} V_2 - V_1&= 523.33- 113.04\\ &=410.29\\ &\approx410.3 \end{align*}$$

\]

Answer:

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