Let's solve the problem step-by-step:
1. Calculate the Volume of the Spherical Container:
- First, we need to determine the volume of the spherical container. The formula for the volume [tex]\( V \)[/tex] of a sphere is given by:
[tex]\[
V = \frac{4}{3} \pi r^3
\][/tex]
- Here, the radius [tex]\( r \)[/tex] is 4 feet.
- Substituting the value of [tex]\( r \)[/tex] into the formula:
[tex]\[
V = \frac{4}{3} \pi (4)^3
\][/tex]
[tex]\[
V = \frac{4}{3} \pi \cdot 64
\][/tex]
[tex]\[
V \approx 268.08 \text{ cubic feet}
\][/tex]
2. Determine the Area of the Base of the Rectangular Tub:
- The area [tex]\( A \)[/tex] of the base of the rectangular tub can be found using the formula:
[tex]\[
A = \text{length} \times \text{width}
\][/tex]
- Given the length is 11 feet and the width is 5 feet:
[tex]\[
A = 11 \times 5
\][/tex]
[tex]\[
A = 55 \text{ square feet}
\][/tex]
3. Calculate the Depth of the Water in Feet:
- To find the depth [tex]\( d \)[/tex] of the water in the tub, we use the formula:
[tex]\[
d = \frac{V}{A}
\][/tex]
- Substituting the volume of water and the area of the base:
[tex]\[
d = \frac{268.08}{55}
\][/tex]
[tex]\[
d \approx 4.87 \text{ feet}
\][/tex]
4. Convert the Depth from Feet to Inches:
- Since there are 12 inches in a foot, we convert the depth to inches by multiplying by 12:
[tex]\[
\text{Depth in inches} = d \times 12
\][/tex]
- Substituting the value of [tex]\( d \)[/tex]:
[tex]\[
\text{Depth in inches} \approx 4.87 \times 12
\][/tex]
[tex]\[
\text{Depth in inches} \approx 58.49 \text{ inches}
\][/tex]
5. Round to the Nearest Tenth of an Inch:
- We round 58.49 to the nearest tenth of an inch, which results in:
[tex]\[
\text{Depth (rounded)} \approx 58.5 \text{ inches}
\][/tex]
Therefore, the depth of the water in the tub when the spherical container is emptied into it is approximately 58.5 inches. The correct answer is:
D. 58.5
