Problem 123 - Algebra and Trigonometry

Question

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.

Step-by-Step Solution

Answer

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

Step 1: 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.

Step 2: 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.

Step 3: 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.