Problem 55 - Precalculus Essentials

Question

A tower that is 125 feet tall casts a shadow 172 feet long. Find the angle of elevation of the Sun to the nearest degree. (IMAGE CANNOT COPY)

Step-by-Step Solution

Answer

36 degrees

Step 1 Identify the given measurements

We are given a tower of height 125 feet and its shadow of length 172 feet. These form a right triangle, and their measurements can be used to solve for the angle of elevation

Step 2 Use the tangent function

The tangent of an angle in a right triangle is defined as the ratio of the length of the opposite side to the length of the adjacent side. So, \(\tan(\theta) = \frac{Opposite Side}{Adjacent Side}\). In our case, the opposite side is 125 feet (height of the tower) and adjacent side is 172 feet (length of shadow). So we have \(\tan(\theta) = \frac{125}{172}\).

Step 3 Solve for the angle

We will have to find the angle whose tangent is \(\frac{125}{172}\). For doing this, we will use an inverse tangent function or arctan. This can be done using a scientific calculator. So \(\theta = \arctan(\frac{125}{172})\)

Step 4 Round to the nearest degree

The arctan function will give the angle in radians. It needs to be converted into degrees by multiplying by \(\frac{180}{\pi}\). After finding the angle in degrees, use standard rounding rules to round to the nearest whole number. This will be your final answer.