Sphere Surface Area Calculation

This solution explains how to find the surface area of a sphere when the ratio of its volume to surface area is given.

Understanding the Formulas

• Volume of a sphere ($V$): $V = \frac{4}{3}\pi r^3$
• Surface area of a sphere ($A$): $A = 4\pi r^2$
• Here, '$r$' represents the radius of the sphere.

Calculating the Radius

The problem states that the ratio of the volume to the surface area is 7 cm:

$ \frac{V}{A} = 7 \text{ cm} $

Substitute the formulas for volume and surface area:

$ \frac{\frac{4}{3}\pi r^3}{4\pi r^2} = 7 $

Simplify the expression:

$ \frac{4\pi r^3}{3 \times 4\pi r^2} = 7 $

Cancel out common terms ($4\pi r^2$):

$ \frac{r}{3} = 7 $

Solve for the radius ($r$):

$ r = 7 \times 3 $

$ r = 21 \text{ cm} $

Finding the Surface Area

Now, use the calculated radius ($r = 21$ cm) to find the surface area ($A$) using its formula:

$ A = 4\pi r^2 $

Substitute the value of $r$:

$ A = 4\pi (21)^2 $

Calculate $21^2$:

$ A = 4\pi (441) $

Multiply to find the final surface area:

$ A = 1764\pi \text{ cm}^2 $

The surface area of the sphere is $1764\pi \text{ cm}^2$.