Question

Question
Stavros is looking up at a tree that is 42 feet away from him, at an angle of elevation of 23°
. What is the tree’s height, x, and the distance between Stavros and the top of the tree, y? (1 point)
Responses

x=98.95 ft.
and y=107.49 ft.
x equals 98.95 ft. and x equals 107.49 ft.

x=45.63 ft.
and y=17.83 ft.
x equals 45.63 ft. and x equals 17.83 ft.

x=17.83 ft.
and y=45.63 ft.
x equals 17.83 ft. and x equals 45.63 ft.

x=17.83 ft.
and y=38.03 ft.

Answer 1

To solve the problem, we can use trigonometry. The situation describes a right triangle where:

- The distance from Stavros to the base of the tree is the adjacent side of the triangle (42 feet).
- The height of the tree is the opposite side of the triangle (x).
- The angle of elevation to the top of the tree is 23°.

We can use the tangent function:

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

Here, \(\theta\) is 23°, and using \(x\) for the height of the tree, we have:

\[
\tan(23°) = \frac{x}{42}
\]

Now, solving for \(x\):

\[
x = 42 \cdot \tan(23°)
\]

Using a calculator, we find:

\[
\tan(23°) \approx 0.4245
\]

So,

\[
x \approx 42 \cdot 0.4245 \approx 17.83 \text{ ft.}
\]

Next, we calculate the distance \(y\) between Stavros and the top of the tree. To find this hypotenuse, we can use the cosine function:

\[
\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}
\]

Rearranging gives us:

\[
\text{hypotenuse} = \frac{\text{adjacent}}{\cos(\theta)} = \frac{42}{\cos(23°)}
\]

Now calculating \(\cos(23°)\):

\[
\cos(23°) \approx 0.9205
\]

Then,

\[
y \approx \frac{42}{0.9205} \approx 45.63 \text{ ft.}
\]

Putting it all together:

- The height of the tree \(x\) is approximately 17.83 feet.
- The distance to the top of the tree \(y\) is approximately 45.63 feet.

Thus, the correct answer is:

**x = 17.83 ft. and y = 45.63 ft.**