Sphere 1 has surface area A1 and volume V1, and sphere 2 has surface area A2 and volume V2. If the radius of sphere 2 is 9 times the radius of sphere 1, what is the ratio of each of the following?
(a) the areas, A2/A1

Given that the radius of sphere 2 (\( r_2 \)) is 9 times the radius of sphere 1 (\( r_1 \)), we need to find the ratio of their surface areas, \( \frac{A_2}{A_1} \).

### Surface Area of a Sphere

The surface area \( A \) of a sphere is given by:

\[
A = 4\pi r^2
\]

Where \( r \) is the radius of the sphere.

### Ratio of the Surface Areas

For sphere 1, the surface area is:

\[
A_1 = 4\pi r_1^2
\]

For sphere 2, the surface area is:

\[
A_2 = 4\pi r_2^2
\]

Given that \( r_2 = 9r_1 \), substitute \( r_2 \) into the equation for \( A_2 \):

\[
A_2 = 4\pi (9r_1)^2 = 4\pi \times 81r_1^2 = 81 \times 4\pi r_1^2
\]

Now, the ratio of the surface areas \( \frac{A_2}{A_1} \) is:

\[
\frac{A_2}{A_1} = \frac{81 \times 4\pi r_1^2}{4\pi r_1^2} = 81
\]

So, the ratio of the areas \( \frac{A_2}{A_1} \) is \( \boxed{81} \).

Would you like details on the other ratios or have any questions?

Here are five related questions:
1. What is the ratio of the volumes \( \frac{V_2}{V_1} \) for the same spheres?
2. How does the ratio of the radii affect the volume ratio of two spheres?
3. What is the formula for the volume of a sphere?
4. If the radius of a sphere is doubled, by what factor does its surface area increase?
5. What is the relationship between the surface area and volume of a sphere?

**Tip:** The surface area of a sphere scales with the square of the radius, while the volume scales with the cube of the radius.

Explore the ratio of surface areas between two spheres where one sphere's radius is 9 times the radius of the other. Understand how surface area scales with the square of the radius using detailed calculations and formulas. Suitable for high school students studying geometry and spheres.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation