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=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. x equals 17.83 ft.  and  x equals 38.03 ft. x=98.95 ft.  and y=107.49 ft.

Answer 1

To find the height of the tree (x) and the distance from Stavros to the top of the tree (y), we can use trigonometry.

1. 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 Stavros to the base of the tree): \[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \implies \tan(23^\circ) = \frac{x}{42} \]

   Rearranging to find \(x\): \[ x = 42 \cdot \tan(23^\circ) \] Let's calculate it: \[ \tan(23^\circ) \approx 0.4245 \] \[ x = 42 \cdot 0.4245 \approx 17.83 \text{ ft} \]

2. Now, to find \(y\) (the distance from Stavros to the top of the tree), we can use the Pythagorean theorem: \[ y = \sqrt{x^2 + 42^2} \] Substituting \(x\): \[ y = \sqrt{(17.83)^2 + 42^2} = \sqrt{318.0289 + 1764} = \sqrt{2082.0289} \approx 45.6 \text{ ft} \]

Thus, the results are:

• Height of the tree \(x \approx 17.83 \text{ ft}\)
• Distance to the top of the tree \(y \approx 45.6 \text{ ft}\)

So the correct answer is:

• \(x = 17.83 \text{ ft}\) and \(y \approx 45.63 \text{ ft}\)

Therefore, the correct response would be: x = 17.83 ft. and y = 45.63 ft.