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Chapter 5: Problem 123

Assuming Earth to be a sphere of radius 4000 miles, how many miles north of the Equator is Miami, Florida, if it is \(26^{\circ}\) north from the Equator? Round your answer to the nearest mile.

Short Answer

Expert verified
Miami, Florida is approximately 1822 miles north of the Equator.

Step by step solution

01

Understand the problem

We know that in a circle, the distance around the edge (the circumference) forms a \(360^{\circ}\) angle. Thus, a \(1^{\circ}\) angle represents \(\frac{1}{360}\) of the circle's circumference. Now, to find the distance from the Equator to Miami, we need to find the distance of the \(26^{\circ}\) arc from the circumference.
02

Calculate the circumference of the Earth

The formula for the circumference of a sphere is \(2\pi r\) where \(r\) is the radius. Substituting \(r = 4000\) miles, we get \(2\pi \times 4000 = 8000\pi\) miles.
03

Calculate the distance from Equator to Miami

We know that \(1^{\circ}\) makes up \(\frac{1}{360}\) of the circumference. Extending this, a \(26^{\circ}\) arc will represent \(\frac{26}{360}\) of the circumference. Applying this to our calculated circumference gives us the distance from the Equator to Miami as \( \frac{26}{360} \times 8000\pi \approx 1822 \) miles.

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