The problem states that the angle of elevation to a tree from a point on the ground is $39^{\circ}$. The distance on the ground from the point to the bottom of the tree is 46 feet. We need to find the height of the tree, rounded to the nearest hundredth of a foot.

1. Problem Description

The problem states that the angle of elevation to a tree from a point on the ground is $39^{\circ}$. The distance on the ground from the point to the bottom of the tree is 46 feet. We need to find the height of the tree, rounded to the nearest hundredth of a foot.

2. Solution Steps

We have a right triangle where the distance from the point on the ground to the base of the tree is the adjacent side to the angle of elevation, and the height of the tree is the opposite side. We can use the tangent function to relate the angle, the adjacent side, and the opposite side.

The formula is:

$tan(\theta) = \frac{opposite}{adjacent}$

In our case, $\theta = 39^{\circ}$, the adjacent side is 46 feet, and the opposite side is the height of the tree, which we will call $x$.

$tan(39^{\circ}) = \frac{x}{46}$

To solve for $x$, we multiply both sides by 46:

$x = 46 \cdot tan(39^{\circ})$

Using a calculator, we find that $tan(39^{\circ}) \approx 0.809784033$

$x = 46 \cdot 0.809784033$

$x \approx 37.250065518$

Rounding to the nearest hundredth of a foot, we have $x \approx 37.25$.

3. Final Answer

The height of the tree is approximately 37.25 feet.