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]
