Given that [tex]\(\frac{3}{8}\)[/tex] of the surface area of a sphere is [tex]\(75\pi \, \text{cm}^2\)[/tex], we need to find the diameter of the sphere. We will break this problem into several steps.
1. Determine the Total Surface Area:
We know that:
[tex]\[
\frac{3}{8} \times \text{Surface Area} = 75\pi
\][/tex]
To find the total surface area of the sphere, we solve for [tex]\(\text{Surface Area}\)[/tex]:
[tex]\[
\text{Surface Area} = \frac{75\pi}{\frac{3}{8}} = 75\pi \times \frac{8}{3} = 200\pi \, \text{cm}^2
\][/tex]
2. Relate Surface Area to the Radius:
The formula for the surface area of a sphere is given by:
[tex]\[
\text{Surface Area} = 4\pi r^2
\][/tex]
Therefore, we set up the following equation:
[tex]\[
4\pi r^2 = 200\pi
\][/tex]
3. Solve for [tex]\(r^2\)[/tex]:
By dividing both sides of the equation by [tex]\(4\pi\)[/tex], we get:
[tex]\[
r^2 = \frac{200\pi}{4\pi} = \frac{200}{4} = 50
\][/tex]
4. Find the Radius [tex]\(r\)[/tex]:
Solving for [tex]\(r\)[/tex], we get:
[tex]\[
r = \sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2} \, \text{cm}
\][/tex]
5. Calculate the Diameter:
The diameter [tex]\(d\)[/tex] of the sphere is twice the radius:
[tex]\[
d = 2r = 2 \times 5\sqrt{2} = 10\sqrt{2} \, \text{cm}
\][/tex]
Therefore, the diameter of the sphere is [tex]\(10\sqrt{2} \, \text{cm}\)[/tex], where [tex]\(a = 10\)[/tex] and [tex]\(b = 2\)[/tex].
In the given form, the diameter is [tex]\(10\sqrt{2}\)[/tex], with [tex]\(a = 10\)[/tex] and [tex]\(b = 2\)[/tex], where [tex]\(b\)[/tex] is a prime number.
