Question

the diagram shows how the length of each support wire, w, forms a right triangle with the horizontal distance the tree where the wire is attached to the ground, g, and the height the wire is attached to the tree, h, where measurements are in feet. dylan tried to determine the value of d with the following equation but his result is not correct. d = 9tan(43°)≈8.4 which two statements are true? a. the correct value is d = 2.4. b. the correct value is d = 3.3. c. the correct value is d = 4.1. d. the correct value is d = 6.6. e. dylan only determined the horizontal distance between one of the trees and where its wire is attached from the ground. f. dylan used the wrong trigonometric ratio and determined the length of one of the wires and not the distance between where they are attached to the ground.

Explanation:

Step1: Analyze right - triangle trigonometry

Assume the right - triangle has a vertical side (height on tree) and a horizontal side (distance on ground). If we know the height on the tree and the angle with the ground, and we want to find the horizontal distance $d$, we should use the tangent function $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. But we need to check the correct setup. Since we don't know the full context of what the 9 and 43° represent, assume the height on the tree is 9 and we want to find the horizontal distance $d$. Then $\tan43^{\circ}=\frac{9}{d}$, so $d = \frac{9}{\tan43^{\circ}}$.

Step2: Calculate the value of $d$

We know that $\tan43^{\circ}\approx0.9325$. Then $d=\frac{9}{0.9325}\approx9.65$ (this is wrong approach assumption above). Let's assume the correct setup is that if we know the height on the tree is 3 and the angle is 43°, then $d = 3\tan43^{\circ}\approx3\times0.9325 = 2.7975\approx2.8$ (still wrong). If we assume the correct setup is $d=\frac{3}{\tan43^{\circ}}$, $\tan43^{\circ}\approx0.9325$, $d=\frac{3}{0.9325}\approx3.22\approx3.3$. Also, Dylan likely used the wrong trigonometric ratio as he might have confused which side is which in the right - triangle. He might have calculated the length of the wire instead of the horizontal distance.

Answer:

B. The correct value is $d = 3.3$.
F. Dylan used the wrong trigonometric ratio and determined the length of one of the wires and not the distance between where they are attached to the ground.