To find the diameter of a hemisphere with a volume of 5990 cubic inches, we'll follow these steps using mathematical formulas:
### Step-by-Step Solution
1. Identify the formula for the volume of a hemisphere:
The volume [tex]\( V \)[/tex] of a hemisphere is given by:
[tex]\[
V = \frac{2}{3} \pi r^3
\][/tex]
where [tex]\( r \)[/tex] is the radius of the hemisphere.
2. Rearrange the formula to solve for [tex]\( r^3 \)[/tex]:
To isolate [tex]\( r^3 \)[/tex], we multiply both sides by [tex]\(\frac{3}{2}\)[/tex]:
[tex]\[
r^3 = \frac{3 \cdot V}{2 \pi}
\][/tex]
3. Substitute the given volume into the formula:
[tex]\[
r^3 = \frac{3 \cdot 5990}{2 \pi}
\][/tex]
Using the value of [tex]\(\pi \approx 3.14159\)[/tex], we get:
[tex]\[
r^3 = \frac{17970}{2 \cdot 3.14159} \approx \frac{17970}{6.28318} \approx 2859.79
\][/tex]
4. Solve for [tex]\( r \)[/tex]:
To find the radius [tex]\( r \)[/tex], we take the cube root of [tex]\( r^3 \)[/tex]:
[tex]\[
r = \sqrt[3]{2859.79} \approx 14.2 \text{ inches}
\][/tex]
5. Determine the diameter:
The diameter [tex]\( D \)[/tex] of the hemisphere is twice the radius:
[tex]\[
D = 2r = 2 \times 14.2 \approx 28.4 \text{ inches}
\][/tex]
6. Round the diameter to the nearest tenth:
The calculated diameter is [tex]\( 28.4 \)[/tex] inches, which is already rounded to the nearest tenth.
### Final Answer
The diameter of the hemisphere, rounded to the nearest tenth of an inch, is [tex]\( \boxed{28.4} \)[/tex] inches.
