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.