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?

Answer 1

recall that, there are 12 inches in 1 foot, now, when the ramp is 4 feet away, that's 4*12 inches, and when it's 6 feet away, that's 6*12 inches.

$\bf \begin{array}{ccll} \stackrel{\textit{inches from driveway}}{x}&\stackrel{\textit{inches of ramp's height}}{y}\\ \text{\textemdash\textemdash\textemdash}&\text{\textemdash\textemdash\textemdash}\\ 48&12\\ 72&18 \end{array}\\\\ -------------------------------$

$\bf \begin{array}{lllll} &x_1&y_1&x_2&y_2\\ % (a,b) &({{ 48}}\quad ,&{{ 12}})\quad % (c,d) &({{ 72}}\quad ,&{{ 18}}) \end{array} \\\\\\ % slope = m \stackrel{\textit{average rate of change}}{slope = {{ m}}= \cfrac{rise}{run}} \implies \cfrac{{{ y_2}}-{{ y_1}}}{{{ x_2}}-{{ x_1}}}\implies \cfrac{18-12}{72-48}\implies \cfrac{6}{24}\implies \cfrac{1}{4}$

Answer 2

Answer:

The rate of change of the ramp's incline is 3 inches up per foot across.

Step-by-step explanation:

Given,

At a point 4 feet from the driveway, the height of the ramp is 12 inches,

The position of a point on ramp = (4,12),

Also, At a point 6 feet from the driveway, the height of the ramp is 18 inches.

The position of another point on the ramp = (6,18),

After making the diagram,

The rate of change of the ramp’s incline = $\frac{18-12}{6-4}$

$=\frac{6}{2}$

$=3\text{ inches up per foot across}$