Question
A child lies on the ground and looks up at the top of a 14-ft tree nearby. The child is 7 ft away from the tree. What is the angle of elevation from the child to the top of the tree? Round to the nearest whole degree.
Answers
GPT-4o mini
To find the angle of elevation from the child to the top of the tree, we can use the tangent function, which relates the opposite side (the height of the tree) to the adjacent side (the distance from the child to the tree).
1. **Identify the sides:**
- Height of the tree (opposite side) = 14 ft
- Distance from the child to the tree (adjacent side) = 7 ft
2. **Use the tangent function:**
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{14 \text{ ft}}{7 \text{ ft}} = 2
\]
3. **Find the angle using the arctangent function:**
\[
\theta = \tan^{-1}(2)
\]
4. **Calculate \(\theta\):**
Using a calculator to find \(\tan^{-1}(2)\):
\[
\theta \approx 63.43^\circ
\]
5. **Round to the nearest whole degree:**
\[
\theta \approx 63^\circ
\]
Thus, the angle of elevation from the child to the top of the tree is approximately **63 degrees**.
1. **Identify the sides:**
- Height of the tree (opposite side) = 14 ft
- Distance from the child to the tree (adjacent side) = 7 ft
2. **Use the tangent function:**
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{14 \text{ ft}}{7 \text{ ft}} = 2
\]
3. **Find the angle using the arctangent function:**
\[
\theta = \tan^{-1}(2)
\]
4. **Calculate \(\theta\):**
Using a calculator to find \(\tan^{-1}(2)\):
\[
\theta \approx 63.43^\circ
\]
5. **Round to the nearest whole degree:**
\[
\theta \approx 63^\circ
\]
Thus, the angle of elevation from the child to the top of the tree is approximately **63 degrees**.