Question

the angle of elevation to a nearby tree from a point on the ground is measured to be 42°. how tall is the tree if the point on the ground is 43 feet from the bottom of the tree? round your answer to the nearest tenth of a foot if necessary.

Explanation:

Step1: Set up tangent - ratio

We have a right - triangle where the angle of elevation is $42^{\circ}$, the adjacent side to the angle is 43 feet, and the opposite side is the height of the tree $x$. The tangent of an angle in a right - triangle is defined as $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. So, $\tan(42^{\circ})=\frac{x}{43}$.

Step2: Solve for $x$

Multiply both sides of the equation by 43: $x = 43\times\tan(42^{\circ})$.
We know that $\tan(42^{\circ})\approx0.9004$. Then $x = 43\times0.9004=38.7172$.

Step3: Round the answer

Rounding $38.7172$ to the nearest tenth, we get $x\approx38.7$.

Answer:

$38.7$ feet