Question

the angle of elevation to a nearby tree from a point on the ground is measured to be 64°. how tall is the tree if the point on the ground is 26 feet from the bottom of the tree? round your answer to the nearest tenth of a foot if necessary.

Explanation:

Step1: Identify the trigonometric relationship

We have a right triangle where the adjacent side to the angle of elevation ($64^\circ$) is 26 feet (distance from the point to the bottom of the tree), and the opposite side is the height of the tree ($x$). We use the tangent function, which is defined as $\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}$. So, $\tan(64^\circ)=\frac{x}{26}$.

Step2: Solve for $x$

To find $x$, we multiply both sides of the equation by 26: $x = 26\times\tan(64^\circ)$. We know that $\tan(64^\circ)\approx2.0503$ (using a calculator). Then, $x = 26\times2.0503\approx53.3078$. Rounding to the nearest tenth, we get $x\approx53.3$.

Answer:

The height of the tree is approximately 53.3 feet.