Problem 54 - Precalculus Essentials

Question

At a certain time of day, the angle of elevation of the Sun is \(40^{\circ} .\) To the nearest foot, find the height of a tree whose shadow is 35 feet long. (IMAGE CANNOT COPY)

Step-by-Step Solution

Answer

The height of the tree to the nearest foot is approximately 29 feet.

Step 1 - Identify relevant sides in the right triangle using the trigonometic relationship

Considering the given problem as a right triangle problem, define the sides in relation to the given angle, \(40^{\circ}\). Here, the shadow of the tree is the 'adjacent side' and the tree's height is the 'opposite side'.

Step 2 - Apply the tangent function

Since we are given the adjacent side (tree shadow) and we have to find the opposite side (tree height), we can use the tangent function, which is the ratio of the opposite side to the adjacent side, apply the formula: \[ \tan(\theta) = \frac{opposite}{adjacent}\] where \(\theta\) is the given angle. Substituting in the given values we get: \[\tan(40^{\circ}) = \frac{tree \ height}{35 feet}\]

Step 3 - Solve for the unknown

To solve for the tree height, rearrange the formula and get the 'tree height' alone on one side by multiplying both sides by 35 feet, then plug in the value of \(\tan(40^{\circ})\) : \[tree \ height = 35 feet * \tan(40^{\circ})\] Solve it to find the height of the tree to the nearest foot.