Question

7 practice 7 (from unit 4, lesson 1)
a step has a height of 6 inches. a ramp starts 5 feet away from the base of the step, making a 5.7° angle with the ground. what can you say about the angle the ramp would make with the ground if the ramp starts closer to the step?
a the angle would decrease.
b the angle would increase.
c the angle would stay the same.
d we cannot determine anything about the angle.

Explanation:

We can think of this as a right - triangle problem. The height of the step is the opposite side of the angle with the ground, and the distance from the base of the step is the adjacent side. The tangent of the angle \(\theta\) with the ground is given by \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). The height of the step (opposite side) is fixed (\(6\) inches). When the ramp starts closer to the step, the adjacent side (distance from the base of the step) decreases. Since \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\) and the opposite side is constant, if the adjacent side \(x\) decreases (\(x_1>x_2>0\)), and \(\tan\theta_1 = \frac{h}{x_1}\), \(\tan\theta_2=\frac{h}{x_2}\) (\(h\) is the height of the step). Because \(y = \tan\theta\) is an increasing function for \(\theta\in(0,\frac{\pi}{2})\), and \(\frac{h}{x_1}<\frac{h}{x_2}\) (since \(x_1 > x_2\)), so \(\theta_1<\theta_2\).

Answer:

B. The angle would increase.