To solve this problem, we need to understand how to use the tangent ratio. The tangent ratio in trigonometry relates the angle of a right triangle to the lengths of the opposite side and the adjacent side. The tangent of an angle [tex]\( \theta \)[/tex] is defined as:
[tex]\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\][/tex]
In this problem:
- The angle [tex]\( \theta \)[/tex] is [tex]\(32^{\circ}\)[/tex].
- The adjacent side to this angle is the distance Alma walked from the tree, which is 125 feet.
- The opposite side is the height of the tree, which we need to find.
Given [tex]\(\theta = 32^{\circ}\)[/tex], let's denote the height of the tree as [tex]\(h\)[/tex]. Applying the tangent ratio, we have:
[tex]\[
\tan(32^{\circ}) = \frac{h}{125}
\][/tex]
From this equation, we can solve for [tex]\(h\)[/tex] by multiplying both sides by 125:
[tex]\[
h = 125 \cdot \tan(32^{\circ})
\][/tex]
Among the given options, Option B:
[tex]\[
\tan 32^{\circ}=\frac{h}{125}
\][/tex]
is the correct choice as it correctly sets up the tangent ratio for solving for the height [tex]\(h\)[/tex] of the tree.
Therefore, the correct tangent ratio to use for finding the height of the tree is:
[tex]\[
\tan 32^{\circ}=\frac{h}{125}
\][/tex]
And the height of the tree is approximately 78 feet when rounded to the nearest foot.
