Question

problem #2:
a tree casts a shadow that is 125 feet in length. the angle of elevation with the sun is 32°. what is the height of the tree?
(there is a right triangle image with one leg 125 ft, angle 32° at the base, and a tree beside it)

Explanation:

Step1: Identify the trigonometric ratio

We have a right triangle where the adjacent side to the \(32^\circ\) angle is the length of the shadow (\(125\) ft) and the opposite side is the height of the tree (\(h\)). The tangent function relates the opposite and adjacent sides: \(\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}\).

Step2: Set up the equation

Substitute \(\theta = 32^\circ\), adjacent \(= 125\) ft, and opposite \(= h\) into the tangent formula: \(\tan(32^\circ)=\frac{h}{125}\).

Step3: Solve for \(h\)

Multiply both sides by \(125\) to isolate \(h\): \(h = 125\times\tan(32^\circ)\). Using a calculator, \(\tan(32^\circ)\approx0.6249\), so \(h\approx125\times0.6249 = 78.1125\) feet.

Answer:

The height of the tree is approximately \(78.1\) feet (or \(78.11\) feet depending on calculator precision).