When we study relationships, we find that there are also inverse square variations. The weight of a body varies inversely as the square of its distance from the center of the earth.

If the radius of the earth is 4000 miles, how much would a 190-pound astronaut orbiting 1000 miles above the surface of the earth weigh?

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Hint

Hint

The astronaut weighs 190 pounds on the surface of the earth which is 4000 miles from its center. Use this fact to find the constant of variation and write a function rule to model the situation.

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Answer

Use your notes to answer the following.

First, find the constant of variation:

W = k over d squared k d²

190 = k over four thousand squared k 4000²

k = 190(4000²) = 3,040,000,000

Next, write a model for the situation.

W = three billion forty million over d squared 3,040,000,000 d²

Substitute in the distance that the astronaut is from Earth – 4000 miles plus 1000 miles.

W = three billion forty million over five thousand squared 3,040,000,000 5000² = 121.6

While orbiting 1000 miles above Earth, the astronaut weighs 121.6 pounds.

Using your notes, describe the relationship between the independent and dependent variables in both a direct and an inverse variation.

1. As the independent variable increases, what happens to the dependent variable?
2. What special features exist with a direct variation and with an inverse variation? (function, family, intercepts, if any, asymptotes, if any, etc.)
3. Write the general equations for a direct variation, a direct square variation, an inverse variation, and an inverse square variation. What is k?
4. Describe one real-world situation for each of the four types of variations listed above.