Problem 56 - Precalculus Essentials

Question

The Washington Monument is 555 feet high. If you are standing one quarter of a mile, or 1320 feet, from the base of the monument and looking to the top, find the angle of elevation to the nearest degree. (IMAGE CANNOT COPY)

Step-by-Step Solution

Answer

The angle of elevation when looking from the base to the top of the Washington Monument is around \(22^\circ\) when rounded to the nearest degree.

Step 1: Identify the known values

The height of the Washington Monument, which is the 'opposite side' in our triangle, is 555 feet. The distance from the base of the monument, the 'adjacent side', is given as a quarter of a mile, or 1320 feet.

Step 2: Use the tangent trigonometric 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. In this case, \(\tan(\Theta) = \frac{opposite}{adjacent} = \frac{555}{1320}\). However, we need to find the angle, not the tangent of the angle.

Step 3: Use the inverse tangent function

To find the angle when we know the tangent of the angle, we use the inverse tangent function, or arctan. Therefore, using a calculator, \(\Theta = \arctan(\frac{opposite}{adjacent}) = \arctan(\frac{555}{1320})\) is evaluated. The calculator should be in degree mode since we want the answer in degrees.

Step 4: Round to the nearest degree

The final answer may not be a whole number, so it needs to be rounded to the nearest degree as per the question.