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Geometrical calculation of planet orbit distances

I could use some help with Algebra Radicals. Thanks!

Radicals

Application Practice

Answer the following questions. Use Equation Editor to write mathematical expressions and equations. First, save this file to your hard drive by selecting Save As from the File menu. Click the white space below each question to maintain proper formatting.

1. For each planet in our solar system, its year is the entire time it takes the planet to revolve once around the sun.
The formula E = 0.3*x^4/3 models the number of Earth days in a planet's year, E, where x is the average distance of the planet from the sun, in millions of kilometers. Use the formula to solve the following problems:

a) We, of course, have 365 Earth days in our year. What is the average distance of the Earth from the sun? Explain how you would solve this problem. Use a calculator and round to the nearest million kilometers.

b) There are approximately 91 Earth days in the year of the planet Mercury. What is the average distance of Mercury from the sun? Explain how you would solve this problem. Use a calculator and round to the nearest million kilometers.

2. The distance to the horizon that you can see, D, in miles from the top of a mountain is given by the formula .
D = Sqrt (2.5h)

a) Solve this equation for h.

b) You've hiked to the top of a mountain with views extending 47 miles to the horizon. How high is the mountain?

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Solution Preview

For each planet in our solar system, its year (in Earth days E ) is the entire time it takes the planet to revolve once around the sun is given as below.

Q1

The formula E = 0.3*x^4/3 (1)

Rearrange

E/0.3 = x^4/3 (2)

Taking Logs of both sides of (2)

Log{E/0.3} = Log{x}^4/3 (3)

Now we use the general Log identity that Log{A}^N ...

Solution Summary

Using a formula to calculate the number of Earth equivalent days that a planet in our solar system takes to orbit the Sun where the number of days E = 0.3*x^4/3 (x being the average distance in millions of km for the planent from the Sun) the average distance of Earth is computed. Also give that the planet Mercury takes 91 Earth days to orbit the Sun its average distance from the Sun is deduced.

The scond part of the question gives a formula for the height of a mointain h seen from a distance D in miles the formula D = Sqrt(2.5h) is used. Given the distance as D = 45 Miles the mountains height is estimated

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