To solve this problem, we need to use trigonometry. The scenario described involves a right triangle, and the tree's height is the vertical leg of that triangle. We are given the following information:

1. From point A, the angle of elevation to the top of the tree is $12.7^\circ$.
2. From point B, the angle between the line AB and the line extending to the tree is $85.4^\circ$.
3. The distance between points A and B is 24.8 feet.

Step 1: Determine the distance from point B to the tree.

We first focus on triangle $ABT$, where:

• $AB = 24.8$ feet,
• $\angle ABT = 85.4^\circ$, and
• $\angle BAT = 12.7^\circ$.

To find the distance $BT$ (from point B to the tree), we can use the law of sines: $\frac{AB}{\sin(\angle BAT)} = \frac{BT}{\sin(\angle ABT)}$ Substitute the known values: $\frac{24.8}{\sin(12.7^\circ)} = \frac{BT}{\sin(85.4^\circ)}$ Solve for $BT$: $BT = \frac{24.8 \cdot \sin(85.4^\circ)}{\sin(12.7^\circ)}$

Step 2: Calculate the height of the tree.

Now that we know $BT$, we can calculate the height of the tree using the angle of elevation from point B and the tangent function. The height of the tree $h$ is: $\tan(12.7^\circ) = \frac{h}{BT}$ Solve for $h$: $h = BT \cdot \tan(12.7^\circ)$

Let’s calculate this step-by-step.The height of the tree is approximately $25.3$ feet.

Would you like more details on the calculation or have any further questions? Here are 5 related questions that expand on this topic:

1. How is the law of sines used in non-right triangles?
2. How does the angle of elevation relate to tangent in right triangles?
3. How can we determine the height of an object using two different points?
4. What are the assumptions when using trigonometry in real-world applications?
5. How would the solution change if the angles were measured in radians?

Tip: When using trigonometry, always check if your calculator is set to the correct mode (degrees or radians) based on the given angles.