Question
Brad is looking up at a tree that is 42 feet away from him, at an angle of elevation of 23 degrees. What is the tree's height, x, and the distance between Brad and the top of the tree, y?
Answers
GPT-4o mini
To find the height of the tree (\(x\)) and the distance between Brad and the top of the tree (\(y\)), we can use trigonometric functions.
1. **Finding the height of the tree (\(x\))**:
We can use the tangent function, which relates the angle of elevation to the opposite side (height of the tree) and the adjacent side (distance from Brad to the tree). The formula is:
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{x}{42}
\]
We can rearrange this to find \(x\):
\[
x = 42 \cdot \tan(23^\circ)
\]
Now, we'll calculate \(x\):
\[
x \approx 42 \cdot 0.4245 \quad (\text{using a calculator for } \tan(23^\circ))
\]
\[
x \approx 17.8 \text{ feet}
\]
2. **Finding the distance between Brad and the top of the tree (\(y\))**:
We can use the cosine function to find the hypotenuse of the right triangle formed:
\[
\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{42}{y}
\]
Rearranging for \(y\):
\[
y = \frac{42}{\cos(23^\circ)}
\]
Now, we'll calculate \(y\):
\[
y \approx \frac{42}{0.9205} \quad (\text{using a calculator for } \cos(23^\circ))
\]
\[
y \approx 45.6 \text{ feet}
\]
So, the height of the tree (\(x\)) is approximately **17.8 feet** and the distance from Brad to the top of the tree (\(y\)) is approximately **45.6 feet**.
1. **Finding the height of the tree (\(x\))**:
We can use the tangent function, which relates the angle of elevation to the opposite side (height of the tree) and the adjacent side (distance from Brad to the tree). The formula is:
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{x}{42}
\]
We can rearrange this to find \(x\):
\[
x = 42 \cdot \tan(23^\circ)
\]
Now, we'll calculate \(x\):
\[
x \approx 42 \cdot 0.4245 \quad (\text{using a calculator for } \tan(23^\circ))
\]
\[
x \approx 17.8 \text{ feet}
\]
2. **Finding the distance between Brad and the top of the tree (\(y\))**:
We can use the cosine function to find the hypotenuse of the right triangle formed:
\[
\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{42}{y}
\]
Rearranging for \(y\):
\[
y = \frac{42}{\cos(23^\circ)}
\]
Now, we'll calculate \(y\):
\[
y \approx \frac{42}{0.9205} \quad (\text{using a calculator for } \cos(23^\circ))
\]
\[
y \approx 45.6 \text{ feet}
\]
So, the height of the tree (\(x\)) is approximately **17.8 feet** and the distance from Brad to the top of the tree (\(y\)) is approximately **45.6 feet**.