Sphere Volume Calculation from Surface Area

The problem asks us to find the volume of a sphere when its surface area is given. We are given the surface area ($A$) and the value of pi ($\pi$) to use.

Given Information:

• Surface Area of the sphere, $A = 2464 \text{ cm}^2$
• Value of pi, $\pi = \frac{22}{7}$

Step 1: Find the radius of the sphere

The formula for the surface area ($A$) of a sphere is:

$$A = 4 \pi r^2$$

Where '$r$' is the radius of the sphere.

We can substitute the given values into the formula:

$$2464 = 4 \times \frac{22}{7} \times r^2$$

Now, let's simplify the equation:

$$2464 = \frac{88}{7} \times r^2$$

To find $r^2$, we rearrange the equation:

$$r^2 = 2464 \times \frac{7}{88}$$

Let's perform the division first: $2464 \div 88 = 28$.

So, the equation becomes:

$$r^2 = 28 \times 7$$

$$r^2 = 196$$

To find the radius '$r$', we take the square root of $196$:

$$r = \sqrt{196}$$

$$r = 14 \text{ cm}$$

So, the radius of the sphere is 14 cm.

Step 2: Calculate the volume of the sphere

The formula for the volume ($V$) of a sphere is:

$$V = \frac{4}{3} \pi r^3$$

Now, substitute the value of the radius ($r = 14 \text{ cm}$) and $\pi = \frac{22}{7}$ into the volume formula:

$$V = \frac{4}{3} \times \frac{22}{7} \times (14)^3$$

First, calculate $14^3$:

$$14^3 = 14 \times 14 \times 14 = 196 \times 14 = 2744$$

Now substitute this back into the volume formula:

$$V = \frac{4}{3} \times \frac{22}{7} \times 2744$$

$$V = \frac{4 \times 22 \times 2744}{3 \times 7}$$

We can simplify by dividing 2744 by 7:

$$2744 \div 7 = 392$$

Now the equation is:

$$V = \frac{4 \times 22 \times 392}{3}$$

Multiply the numbers in the numerator:

$$4 \times 22 = 88$$

$$88 \times 392 = 34496$$

So, the volume is:

$$V = \frac{34496}{3}$$

Finally, perform the division:

$$V \approx 11498.666...$$

Rounding to two decimal places, we get:

$$V \approx 11498.67 \text{ cm}^3$$

Therefore, the volume of the sphere is approximately $11498.67 \text{ cm}^3$.