Question

the diagram shows how the length of each support wire, w, forms a right - triangle with the horizontal distance from the tree where the wire is attached to the ground, g, and the height the wire is attached to the tree, h, where all measurements are in feet. the support wire on one tree was attached at a height of 5.5 feet. the cosine of the angle, θ, the wire made with the tree is 0.72. let α represent the angle the wire makes with the ground. which set of steps shows a correct method to find the length of the wire, w, in feet using a trigonometric function of α? α = 90 - θ sin α = cos θ sin α = 0.72 a. w/5.5 = 0.72 w = 0.72(5.5) w ≈ 4.0 feet α = 90 - θ sin α = cos θ sin α = 0.72 b. 5.5/w = 0.72 5.5 = 0.72w w = 5.5/0.72 w ≈ 7.6 feet α = θ cos α = cos θ cos α = 0.72 c. w/5.5 = 0.72 w = 0.72(5.5) w ≈ 4.0 feet

Explanation:

Step1: Use complementary - angle relationship

Since the angle the wire makes with the tree is $\theta$ and the angle it makes with the ground is $\alpha$, and $\theta+\alpha = 90^{\circ}$, so $\alpha=90 - \theta$. Then, by the co - function identity $\sin\alpha=\cos\theta$. Given $\cos\theta = 0.72$, so $\sin\alpha=0.72$.

Step2: Apply sine function in right - triangle

In the right - triangle formed by the wire, the tree, and the ground, $\sin\alpha=\frac{h}{w}$, where $h = 5.5$ feet (height on the tree where the wire is attached) and $w$ is the length of the wire. Substituting $\sin\alpha = 0.72$ and $h = 5.5$ into $\sin\alpha=\frac{h}{w}$, we get $\frac{5.5}{w}=0.72$.

Step3: Solve for $w$

Cross - multiply to get $5.5 = 0.72w$. Then $w=\frac{5.5}{0.72}\approx7.6$ feet.

Answer:

B. $\frac{5.5}{w}=0.72$, $5.5 = 0.72w$, $w=\frac{5.5}{0.72}$, $w\approx7.6$ feet