Question

the angle of elevation to a nearby tree from a point on the ground is measured to be 54°. how tall is the tree if the point on the ground is 52 feet from the bottom of the tree? round your answer to the nearest hundredth of a foot if necessary.

Explanation:

Step1: Set up the trigonometric relationship

We know that in a right - triangle (where the tree is the vertical side, the distance on the ground is the adjacent side, and the line of sight is the hypotenuse), the tangent of an angle of elevation $\theta$ is given by $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. Let $h$ be the height of the tree. Here, $\theta = 54^{\circ}$ and the adjacent side $x = 52$ feet. So, $\tan(54^{\circ})=\frac{h}{52}$.

Step2: Solve for $h$

We can rewrite the equation as $h = 52\times\tan(54^{\circ})$. Since $\tan(54^{\circ})\approx1.37638192047$ (a known trigonometric value). Then $h=52\times1.37638192047$.

Step3: Calculate the value of $h$

$h = 52\times1.37638192047=71.5718698644$.

Answer:

$71.57$ feet