https://file.aiquickdraw.com/mathbot/file/1726248303768b45qfmlf.jpg

To solve this problem, we will use the formula for the volume of a sphere:

\[
V = \frac{4}{3} \pi r^3
\]

Given that the radius \( r \) of the sphere is \( 12x \, \text{mm} \), we can substitute \( r = 12x \) into the formula for the volume.

1. Substitute \( r = 12x \) into the formula:

\[
V = \frac{4}{3} \pi (12x)^3
\]

2. Now calculate \( (12x)^3 \):

\[
(12x)^3 = 12^3 \times x^3 = 1728x^3
\]

3. Substitute this back into the formula for the volume:

\[
V = \frac{4}{3} \pi \times 1728x^3
\]

4. Multiply \( \frac{4}{3} \) by 1728:

\[
V = \frac{4 \times 1728}{3} \pi x^3 = \frac{6912}{3} \pi x^3 = 2304 \pi x^3
\]

So, the expression for the volume of the sphere in terms of \( \pi \) and \( x \) is:

\[
V = 2304 \pi x^3
\]

Would you like more details or have any questions?

Here are 5 related questions for practice:
1. What would be the volume if the radius were \( 6x \) mm instead of \( 12x \) mm?
2. How does the volume of a sphere change if the radius is doubled?
3. What is the surface area of a sphere with radius \( 12x \) mm?
4. If \( x = 1 \), what is the volume of the sphere in cubic millimeters?
5. How would the formula change if the radius were given in centimeters instead of millimeters?

**Tip:** When scaling the radius of a sphere, the volume increases with the cube of the radius. So doubling the radius results in 8 times the volume!

A sphere has a radius of 12x mm. Write an expression for the volume of the sphere in terms of π and x. Give your answer in its simplest form.

Learn how to find the volume of a sphere with radius 12x mm using the formula V = 4/3 π r^3. This problem covers key geometric concepts and algebraic manipulation. Suitable for students in Grades 8-10.

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