The surface area of a sphere is 275.56 cm². What is the approximate volume of the sphere? A. 2,287.15 π cm³ B. 762.38 π cm³ C. 6,099.06 π cm³ D. 381.19 cm³

First, let's recall the formula for the surface area of a sphere, which is \( 4\pi r^2 \), where \( r \) is the radius of the sphere. We are given the surface area \( A \) as \( 275.56 \text{ cm}^2 \).

We can solve for the radius \( r \) using the formula for surface area: \( A = 4\pi r^2 \)

So, we have \( 275.56 = 4\pi r^2 \)

Divide both sides by \( 4\pi \) to solve for \( r^2 \): \( r^2 = \frac{275.56}{4\pi} \)

Let's calculate it approximately: \( r^2 ≈ \frac{275.56}{4 \times 3.14159} \) \( r^2 ≈ \frac{275.56}{12.56636} \) \( r^2 ≈ 21.92 \)

Now, the square root of \( r^2 \) will give us the radius \( r \): \( r ≈ \sqrt{21.92} \) \( r ≈ 4.68 \) cm

Now we can use the radius to find the volume of the sphere using the formula for the volume of a sphere, which is \( \frac{4}{3} \pi r^3 \): \( V = \frac{4}{3}\pi r^3 \)

Using our calculated radius: \( V ≈ \frac{4}{3}\pi (4.68)^3 \)

Now, calculate the volume approximately: \( V ≈ \frac{4}{3}\pi (102.1664) \) \( V ≈ 1.3333 \pi \times 102.1664 \) \( V ≈ 136.2216 \pi \)

The \(\pi\) in the options indicates that the answer is kept in terms of \( \pi \), so we don't multiply with the value of \( \pi \). Let's match our result with one of the given options:

\( V ≈ 136.2216 \pi \) cm³

The closest match among the options is:

B. 762.38 π cm³

However, the volume we calculated, \( V ≈ 136.2216 \pi \) cm³, does not exactly match option B. It seems there might be a discrepancy in the calculations or a possible rounding issue with the provided options. If we consider the nearest integer, then option B would be the closest, but if precise calculation matters, none of the options accurately reflects the computed volume.