Solve each problem.

Solar Eclipse on Earth The sun has a diame...

Question

Solve each problem. 
 
Solar Eclipse on Earth The sun has a diameter of about 865,000 mi with a maximum distance from Earth's surface of about 94,5​​00,000 mi. The moon has a smaller diameter of 2159 mi. For a total solar eclipse to occur, the moon must pass between Earth and the sun. The moon must also be close enough to Earth for the moon's umbra (shadow) to reach the surface of Earth. (Data from Karttunen, H., P. Kröger, H. Oja, M. Putannen, and K. Donners, Editors, Fundamental Astronomy, Fourth Edition, Springer-Verlag.)

a. Calculate the maximum distance, to the nearest thousand miles, that the moon can be from Earth and still have a total solar eclipse occur. (Hint: Use similar triangles.)

Accepted Answer

Hello, everyone. We're asked to use the concept of similar triangles to solve for the unknown length X. We will round the answer to the nearest mile. When necessary, we are given a triangle with a base of 1980 miles. A top leg of 3726 miles. And within, there is a smaller triangle with a base of 762 miles. And the top leg being the value X. Our four answer choices are a 1397 miles. B 1434 miles. C 1381 miles D 1473 miles. First recall that when we're working with similar triangles, we can set up a proportion. So I'm gonna set up a proportion where my first fraction will be all the information I have about the smaller triangle. And I'm going to set that equal to a fraction with all the information I know about the larger triangle. So in the smaller triangle, I'm going to use my unknown side of X as my numerator and the base of 762 as my denominator. And since I used the top leg as the numerator in my first fraction, that's what I will also do for my larger triangle or my second fraction. So the numerator is gonna be 3726. The denominator will be the base which is 1980. Now to solve this, I need to cross multiply. Sometimes people overheard it referred to as the butterfly method. Either way when I cross multiply X multiplied by 982,980 X and going in the opposite direction, I will multiply 762 by 3726. And when I do so, I get that 1980 X equals 2,839,212. And then to solve for X, I need to divide both sides by 1980. And when I do so, X equals approximately 1433 0.945 miles. Since we are instructed to round to the nearest mile, we will round this up to 1434 miles. That is our answer and it is answer choice B have a nice day.

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