Question

a ramp with a constant incline is made to connect a driveway to a front door. at a point 4 feet from the driveway, the height of the ramp is 12 inches. at a point 6 feet from the driveway, the height of the ramp is 18 inches. what is the rate of change of the ramp’s incline?

\\(\frac{1}{3}\\) inch up per foot across

\\(\frac{1}{2}\\) inch up per foot across

2 inches up per foot across

3 inches up per foot across

Explanation:

Step1: Identify the two points

We have two points: \((x_1, y_1) = (4, 12)\) (where \(x\) is feet from driveway and \(y\) is height in inches) and \((x_2, y_2) = (6, 18)\).

Step2: Use the rate of change formula

The rate of change (slope) is given by \(\frac{y_2 - y_1}{x_2 - x_1}\).
Substitute the values: \(\frac{18 - 12}{6 - 4}=\frac{6}{2} = 3\)? Wait, no, wait. Wait, no, let's check units. Wait, the question is rate of change of incline, which is change in height over change in horizontal distance. Wait, but let's recalculate. Wait, \(y_2 - y_1 = 18 - 12 = 6\) inches, \(x_2 - x_1 = 6 - 4 = 2\) feet. So rate is \(\frac{6\ \text{inches}}{2\ \text{feet}} = 3\) inches per foot? Wait, no, wait, no, wait. Wait, no, wait, the options: wait, let's check again. Wait, at 4 feet, height 12 inches; at 6 feet, height 18 inches. So change in height is \(18 - 12 = 6\) inches, change in horizontal distance is \(6 - 4 = 2\) feet. So rate is \(\frac{6}{2}=3\) inches per foot? Wait, but let's check the options. The last option is 3 inches up per foot across. Wait, but let's verify again. Wait, maybe I made a mistake. Wait, no, let's see: the formula for slope (rate of change) is \(\frac{\Delta y}{\Delta x}\). Here, \(\Delta y = 18 - 12 = 6\) inches, \(\Delta x = 6 - 4 = 2\) feet. So \(\frac{6}{2}=3\) inches per foot. So the rate of change is 3 inches up per foot across.

Answer:

D. 3 inches up per foot across (assuming the options are labeled as A, B, C, D with D being 3 inches up per foot across)