Chapter 1.3, Problem 123AYU

Access Ramp 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.

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

Find and interpret the $x$-intercept of the graph of your equation.

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.

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

To determine

A linear equation that relates the height $y$ of the ramp above the floor to the horizontal distance $x$ from the platform, where 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.

Answer to Problem 123AYU

Solution:

A linear equation that relates the height $y$ of the ramp above the floor to the horizontal distance $x$ from the platform is $y=-\frac{2}{25}x+30$.

Explanation of Solution

Given Information:

A wooden access ramp is being built to reach a platform $30$ inches above the floor.

The ramp drops 2 inches for every $25$- inch run.

Explanation:

The general form linear equation is $y=mx+b$ where $m$ is a slope and $b$ is $y-$ intercept.

By looking the graph, the highest vertical distance is $30$ inches. That is for $x=0$, $y=30$.

Substitute $x=0$, $y=30$ in $y=mx+b$, get

$30=m\cdot 0+b$

$\Rightarrow b=30$

The slope is the change in $x$ divided by change in $y$.

Since, if the ramp drops 2 inches, then change in $y$ is $-2$ and if the run is $25$ inches, then change in $x$ is $25$.

Therefore, slope is $\frac{-2}{25}$.

The linear equation becomes $y=-\frac{2}{25}x+30$.

Therefore, a linear equation that relates the height $y$ of the ramp above the floor to the horizontal distance $x$ from the platform is $y=-\frac{2}{25}x+30$.

To determine

The $x-$ intercept of the graph of equationthat relates the height $y$ of the ramp above the floor to the horizontal distance $x$ from the platform, where 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.

Answer to Problem 123AYU

Solution:

The $x-$ intercept of the graph of equation $y=-\frac{2}{25}x+30$ is $x=375$.

Explanation of Solution

Given Information:

A wooden access ramp is being built to reach a platform $30$ inches above the floor.

The ramp drops 2 inches for every $25$- inch run.

Explanation:

From the part (a), the linear equation is $y=-\frac{2}{25}x+30$.

To find $x-$ intercept set $y=0$.

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

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

Multiply both sides by $-\frac{25}{2}$,

$\Rightarrow x=-30\cdot \left(-\frac{25}{2}\right)$

$\Rightarrow x=15\cdot 25$

$\Rightarrow x=375$

Therefore, the $x-$ intercept of the graph of equation $y=-\frac{2}{25}x+30$ is $x=375$.

To determine

The ramp will meet the requirements or not, where design requirements stipulate that the maximum run be $30$ feet and the maximum slope be a drop of 1 inch for each $12$ inches of run.

Answer to Problem 123AYU

Solution:

It will meet the design requirements for the slope.

Explanation of Solution

Given Information:

A wooden access ramp is being built to reach a platform $30$ inches above the floor.

The ramp drops 2 inches for every $25$- inch run.

Explanation:

Let the maximum run if of the ramp is equal to $30$ feet.

From the part (b), run of the ramp is $375$ inches.

Here, need to convert $375$ inches into feet.

By using $12$ inches = 1 feet.

Implies that, 1 inches = $\frac{1}{12}$ feet.

Implies that, $375$ inches = $\frac{375}{12}$ feet.

$=31.25$ feet

Since, the ramp exceed the maximum run, the ramp will not meet the design requirements.

The slope is division of the change in $y$ by the change in $x$.

Since, if the ramp drops 1 inches, then change in $y$ is $-1$ and if the run is $12$ inches, then change in $x$ is $12$.

Therefore, slope is $\frac{-1}{12}$.

Compare the slopes $\frac{-2}{25}$ and $\frac{-1}{12}$.

It is observed that, ramp slope is less than that of the design requirements.

Since maximum slope can be $\frac{-1}{12}$.

It will meet the design requirements for the slope.

To determine

The slope that could be used to obtain the $30$ inch rise and still meet design requirements.

Answer to Problem 123AYU

Solution:

The slope that could be used to obtain the $30$ inch rise and still meet design requirements is $m=-\frac{1}{12}$.

Explanation of Solution

Given Information:

A wooden access ramp is being built to reach a platform $30$ inches above the floor.

The ramp drops 2 inches for every $25$- inch run.

Explanation:

The maximum run of the ramp cannot larger $30$ feet.

The $x$ intercept of the graph in part (b) is also the run of the ramp in inches.

Convert $30$ feet into inches by using 1 feet =$12$ inches.

Implies that, $30$ feet = $30\cdot 12$ inches = $360$ inches.

The run of the ramp not to be more than $360$ inches based on the given design requirements.

Since, $x$ intercept occur when $y=0$.

Therefore, $x$ intercept of the design is $360$ inches.

To determine slope, let use the given rise of $30$ inches in the standard linear form.

The general form linear equation is $y=mx+b$ where $m$ is a slope and $b$ is $y-$ intercept.

To find $m$, plug $x=360,y=0$, and $b=30$ in $y=mx+b$

$\Rightarrow 0=m\cdot 360+30$

$\Rightarrow -30=360m$

$\Rightarrow m=-\frac{30}{360}$

$\Rightarrow m=-\frac{1}{12}$

Therefore, the slope that could be used to obtain the $30$ inch rise and still meet design requirements is $m=-\frac{1}{12}$.