Question

The angle of elevation to a nearby tree from a point on the ground is measured to be 32degrees
∘
. How tall is the tree if the point on the ground is 71 feet from the bottom of the tree? Round your answer to the nearest tenth of a foot if necessary.

Answer 1

To find the height of the tree, we can use the tangent function from trigonometry. The angle of elevation (32 degrees) and the distance from the point on the ground to the base of the tree (71 feet) form a right triangle.

We use the tangent of the angle of elevation, which is defined as the opposite side (the height of the tree, \( h \)) over the adjacent side (the distance from the tree, 71 feet):

\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{h}{71}
\]

Replacing \( \theta \) with 32 degrees:

\[
\tan(32^\circ) = \frac{h}{71}
\]

Now, we solve for \( h \):

\[
h = 71 \times \tan(32^\circ)
\]

Next, we calculate \( \tan(32^\circ) \) using a calculator:

\[
\tan(32^\circ) \approx 0.6249
\]

Now substituting this value back into the equation for \( h \):

\[
h = 71 \times 0.6249 \approx 44.3999
\]

Finally, rounding to the nearest tenth of a foot gives:

\[
h \approx 44.4 \text{ feet}
\]

Therefore, the height of the tree is approximately **44.4 feet**.