Questions: A tree casts a 25-foot shadow on a sunny day, as shown in the diagram below. If the angle of elevation from the tip of the shadow to the top of the tree is 32 degrees, what is the height of the tree to the nearest tenth of a foot?

Solution Steps

We are given the angle of elevation, the length of the shadow (adjacent side), and we need to find the height of the tree (opposite side). The trigonometric ratio that relates the opposite side and the adjacent side is the tangent function:

\\(\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\\)

In our case, the angle of elevation \\(\theta\\) is 32°, the adjacent side is 25 feet (the shadow), and the opposite side is the height of the tree (h). So we can set up the equation:

\\(\tan(32^{\circ}) = \frac{h}{25}\\)

Multiply both sides of the equation by 25:

\\(h = 25 \times \tan(32^{\circ})\\)

Now, use a calculator to find the value of \\(\tan(32^{\circ})\\):

\\(\tan(32^{\circ}) \approx 0.6249\\)

Substitute this value into the equation:

\\(h = 25 \times 0.6249\\) \\(h \approx 15.6225\\)

Round the height to the nearest tenth of a foot:

\\(h \approx 15.6\\) feet

Final Answer