To find the diameter of a sphere given its volume, we need to follow several steps involving the volume formula for a sphere. Let's work through the problem step-by-step.
1. Understand the Volume Formula:
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.
2. Given Information:
We know the volume of the sphere is [tex]\(2304 \pi \, \text{mm}^3\)[/tex].
3. Set Up the Equation:
Substitute the given volume back into the volume formula:
[tex]\[
2304 \pi = \frac{4}{3} \pi r^3
\][/tex]
4. Solve for [tex]\(r^3\)[/tex]:
First, we will isolate [tex]\(r^3\)[/tex] by dividing both sides of the equation by [tex]\(\pi\)[/tex]:
[tex]\[
2304 = \frac{4}{3} r^3
\][/tex]
Next, to solve for [tex]\(r^3\)[/tex], multiply both sides by [tex]\(\frac{3}{4}\)[/tex]:
[tex]\[
r^3 = 2304 \times \frac{3}{4} = 1728
\][/tex]
5. Find [tex]\(r\)[/tex]:
To find [tex]\(r\)[/tex], take the cube root of both sides:
[tex]\[
r = \sqrt[3]{1728}
\][/tex]
The cube root of 1728 is approximately 12:
[tex]\[
r \approx 12 \, \text{mm}
\][/tex]
6. Calculate the Diameter:
The diameter [tex]\(d\)[/tex] of the sphere is twice the radius:
[tex]\[
d = 2r = 2 \times 12 = 24 \, \text{mm}
\][/tex]
7. Round to the Nearest Whole Number (if necessary):
In this case, the diameter is already a whole number, so no further rounding is needed.
Thus, the diameter of the sphere is [tex]\(24 \, \text{mm}\)[/tex].
