Q. 123 - Precalculus Enhanced with Graphing Utilities

Question

A wooden access ramp is being built to reach a platform that sits 30 inches above the floor. The ramp drops 2 inches for every 25-inch run.

(a) Write a linear equation that relates the height y of the ramp above the floor to the horizontal distance x from the platform.

(b) Find and interpret the x-intercept of the graph of your equation.

(c) Design requirements stipulate that the maximum run be 30 feet and that the maximum slope be a drop of 1 inch for each 12 inches of run. Will this ramp meet the requirements? Explain.

(d) What slopes could be used to obtain the 30-inch rise and still meet design requirements?

Step-by-Step Solution

Answer

Part (a) The linear equation is: $y=-\frac{2}{25}x+30$

Part (b) x-intercept is 375 inches. It means that the ramp meets the floor 375 inches (31.25 feet) from the base of the platform.

Part (c) The ramp does not meet the requirements as it has a run of 31.25 feet. 

Part (d) The only slope possible for the ramp to comply with the requirement is for it to drop 1 inch for every 12 inches run.

Part (a) Step 1. Given information

A wooden access ramp is being built to reach a platform that sits 30 inches above the floor. The ramp drops 2 inches for every 25-inch run.

Part (a) Step 2. Finding the linear equation

Slope, $m=\frac{rise}{run}$

$m=\frac{-2}{25}$

y-intercept, $b=30$

The required equation is: $y=mx+b$

$y=-\frac{2}{25}x+30$

Part (b) Step 1. Finding and interpreting the x-intercept of the graph

We have, $y=-\frac{2}{25}x+30$

to get x-intercept, put $y=0$ in the above equation.

$0=-\frac{2}{25}x+30$

$\frac{2}{25}x=30$

$x=\frac{30\times 25}{2}$

$x=375$

It means that the ramp meets the floor 375 inches (31.25 feet) from the base of the platform.

Part (c) Step 1. Comparing the design requirements and given ramp

Design requirements stipulate that the maximum run be 30 feet and that the maximum slope be a drop of 1 inch for each 12 inches of run. Therefore, the ramp does not meet the requirements as it has a run of 31.25 feet.

Part (d) Step 1. Finding slopes that could be used to obtain the 30-inch rise and still meet design requirements

The only slope possible for the ramp to comply with the requirement is for it to drop 1 inch for every 12 inches run.