Question

the angle of elevation to a nearby tree from a point on the ground is measured to be $22^{circ}$. how tall is the tree if the point on the ground is 92 feet from the bottom of the tree? round your answer to the nearest hundredth of a foot if necessary.

Explanation:

Step1: Define trigonometric relationship

We use the tangent function, where $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$. Here, $\theta=22^\circ$, opposite side is tree height $x$, adjacent side is 92 ft.
$\tan(22^\circ) = \frac{x}{92}$

Step2: Solve for $x$

Rearrange the formula to isolate $x$:
$x = 92 \times \tan(22^\circ)$
Calculate $\tan(22^\circ) \approx 0.4040$, then:
$x \approx 92 \times 0.4040$

Step3: Compute final value

$x \approx 37.168$
Round to nearest hundredth: $x \approx 37.17$

Answer:

37.17 feet