Question

two ramps are built with the same angle of elevation of 12°. one ramp is twice as tall as the other ramp. the heights of the ramps are h₁ and h₂ and the horizontal distances for the ramps are d₁ and d₂. which statement best explains how the ratios of height to horizontal distance for each ramp compare? a. the ratio is twice as big for the taller ramp because the side lengths of the triangle twice as big. the ratio of the height to horizontal distance in each triangle can also be represented by sin 12°. b. the ratio is twice as big for the taller ramp because the side lengths of the triangle twice as big. the ratio of the height to horizontal distance in each triangle can also be represented by tan 12°. c. the ratio is the same for both ramps because the triangles are similar with proportional side lengths. the ratio of the height to horizontal distance in each triangle can also be represented by sin 12°. d. the ratio is the same for both ramps because the triangles are similar with proportional side lengths. the ratio of the height to horizontal distance in each triangle can also be represented by tan 12°.

Explanation:

Step1: Recall the concept of similar - triangles

Two right - triangles formed by the ramps have the same angle of elevation ($12^{\circ}$). So, they are similar triangles. In similar triangles, the ratios of corresponding side lengths are equal.

Step2: Define the ratio of height to horizontal distance

The ratio of the height ($h$) to the horizontal distance ($d$) of a right - triangle with an angle of elevation $\theta$ is given by the tangent function. That is, $\tan\theta=\frac{h}{d}$. Here, $\theta = 12^{\circ}$, so $\tan12^{\circ}=\frac{h_1}{d_1}=\frac{h_2}{d_2}$.

Answer:

D. The ratio is the same for both ramps because the triangles are similar with proportional side lengths. The ratio of the height to horizontal distance in each triangle can also be represented by $\tan12^{\circ}$.