Question

two photographs of the moon were taken at precisely the same time from two locations. the lunar angles of elevation at city a and at city b were measured as 51.6875° and 51.7455°, respectively. the two cities are 396 kilometers apart. calculate the distance to the moon from city a on this day, and compare it with the actual value of 406,000 kilometers. disregard the curvature of earth in this calculation. select the correct choice below and, if necessary, fill in the answer box to complete your choice. (round to the nearest thousand as needed.) a. the distance to the moon from city a is about □ km, which compares favorably to the actual value. b. the distance to the moon from city a is about □ km, which does not compare favorably to the actual value.

Explanation:

Step1: Find the angle at the moon

The difference in the angles of elevation is the angle at the moon. Let \(\theta = 51.7455^{\circ}-51.6875^{\circ}=0.058^{\circ}\)

Step2: Use the Law of Sines

Let \(d\) be the distance from city \(A\) to the moon. The Law of Sines states that \(\frac{d}{\sin(90^{\circ})}=\frac{396}{\sin(0.058^{\circ})}\)

Since \(\sin(90^{\circ}) = 1\), we have \(d=\frac{396}{\sin(0.058^{\circ})}\)

We know that \(\sin(0.058^{\circ})\approx0.001012\)

So \(d=\frac{396}{0.001012}\approx391304.35\approx391000\) (rounded to the nearest thousand)

Answer:

A. The distance to the moon from city A is about \(391000\) km, which compares favorably to the actual value.