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?
A. [tex]\(\frac{1}{3}\)[/tex] inch up per foot across B. [tex]\(\frac{1}{2}\)[/tex] inch up per foot across C. 2 inches up per foot across D. 3 inches up per foot across
To determine the rate of change of the ramp's incline, we need to calculate the slope of the ramp between the two given points. The points provided are (4 feet, 12 inches) and (6 feet, 18 inches).
We will use the formula for the slope [tex]\( m \)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex]: [tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Let's identify the coordinates of the two points: - [tex]\( (x_1, y_1) = (4, 12) \)[/tex] - [tex]\( (x_2, y_2) = (6, 18) \)[/tex]
Now, let's plug these values into the slope formula: [tex]\[ m = \frac{18 - 12}{6 - 4} \][/tex]
Simplify the numerator and the denominator separately: [tex]\[ 18 - 12 = 6 \][/tex] [tex]\[ 6 - 4 = 2 \][/tex]
So, we have: [tex]\[ m = \frac{6}{2} \][/tex]
Now, divide to find the slope: [tex]\[ m = 3 \][/tex]
Therefore, the rate of change of the ramp's incline is [tex]\( 3 \)[/tex] inches up per foot across.
The correct answer is: [tex]\[ \boxed{3 \text{ inches up per foot across}} \][/tex]
