The angle of elevation to a nearby tree from a point on the ground is measured to be 66degrees∘. How tall is the tree if the point on the ground is 68 feet from the bottom of the tree? Round your answer to the nearest tenth of a foot if necessary.

To determine the height of the tree, we can again use trigonometry, specifically the tangent function. This time, we know the horizontal distance and the angle of elevation, and we want to find the height of the tree.

### Step 1: Identify the known values.
- Horizontal distance from the point on the ground to the base of the tree (adjacent side): \( d = 68 \) feet
- Angle of elevation: \( \theta = 66^\circ \)

### Step 2: Write the equation using the tangent function.
The tangent of an angle in a right triangle is the ratio of the opposite side (height of the tree) to the adjacent side (horizontal distance):

\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{h}{d}
\]

Where:
- \( h \) is the height of the tree (the opposite side).

### Step 3: Solve for the height \( h \).
\[
h = d \times \tan(\theta)
\]

Substituting the known values:

\[
h = 68 \times \tan(66^\circ)
\]

### Step 4: Calculate the value.
Using a calculator:

\[
h \approx 68 \times \tan(66^\circ) \approx 68 \times 2.246 \approx 152.7 \text{ feet}
\]

### Final Answer:
The height of the tree is approximately **152.7 feet**.

Would you like more details or have any questions?

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Here are some related questions to consider:
1. How would the height of the tree change if the distance was reduced to 50 feet?
2. What would be the height if the angle of elevation were increased to 75 degrees?
3. How does the calculation change if the tree is on a hill above the measuring point?
4. How would you calculate the height if the distance and height were both unknown?
5. What is the height of the tree if the angle of elevation was measured as 65 degrees instead of 66 degrees?
6. Can you determine the height using the sine or cosine functions instead of tangent?
7. What would the horizontal distance be if the height was given and the angle was 66 degrees?
8. How would the result differ if the point was above ground level, not on it?

**Tip:** When solving trigonometric problems, ensure the angle is accurate and consider using more decimal places in intermediate steps to improve precision.

Learn how to calculate the height of a tree using trigonometry. Given the angle of elevation and horizontal distance, find the height of the tree using the tangent function. Step-by-step solution provided with example calculations. Suitable for high school students studying trigonometry.

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