Question

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. (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?

Short answer

The linear equation is \[ y = -\frac{2}{25}x + 30 \]. The x-intercept is 375 inches. The ramp does not meet the maximum run but meets the maximum slope requirement. Suitable slopes are between \[ 1/30 \] and \[ 1/12 \].

Step-by-step solution

- Determine the Slope

The problem states that the ramp drops 2 inches for every 25-inch run. Therefore, the slope (m) is calculated as the change in height over the change in horizontal distance: \[ m = \frac{{-2}}{{25}} \] The negative sign indicates a drop.

- Write the Linear Equation

Use the slope-intercept form of a linear equation, which is \[ y = mx + b \] Here, the height at the beginning (the platform) is 30 inches, so the y-intercept (b) is 30. Substituting the slope (m) and y-intercept (b) gives us: \[ y = -\frac{2}{25}x + 30 \]

- Find the x-intercept

The x-intercept occurs where the height (y) is 0. Set the equation to 0 and solve for x: \[ 0 = -\frac{2}{25}x + 30 \] \[ \frac{2}{25}x = 30 \] \[ x = \frac{30 \times 25}{2} \] \[ x = 375 \] This means the horizontal distance (x) from the platform to the floor is 375 inches.

- Check Maximum Run Requirement

Convert the maximum run (30 feet) to inches: \[ 30 \text{ feet} = 30 \times 12 \text{ inches} = 360 \text{ inches} \] Since 375 inches > 360 inches, the maximum run requirement is not satisfied.

- Check Maximum Slope Requirement

The maximum slope allowed is a drop of 1 inch for each 12 inches of run, which is \[ \frac{1}{12} \] The slope of the ramp is \[ \frac{2}{25} \text{ (converted to positive for comparison)} \] Comparing both slopes: \[ \frac{2}{25} \text{ (approximately equal to 0.08)} \text{ is less than } \frac{1}{12} \text{ (approximately equal to 0.083)} \] Therefore, the ramp's slope meets the maximum slope requirement.

- Determine Other Slope Options

To obtain a 30-inch rise while meeting the design requirements (maximum slope drop of 1 inch for every 12 inches of run), the slope needs to be between \[ \frac{1}{30} \text{ and } \frac{1}{12} \] Evaluating these slopes, the minimum horizontal distance for the rise can be calculated as follows: \[ \text{Max distance: } x = 30 \times 12 = 360 \text{ inches} \] \[ \text{Min distance: } x = 30 \times 30 = 900 \text{ inches} \] Therefore, a horizontal distance between 360 inches and 900 inches would meet the slope requirement.

Key concepts

Linear Equations

Linear equations are foundational in algebra and appear frequently in real-life scenarios. In algebra, a linear equation describes a relationship between two variables, where one depends on the other.

Linear equations are written in the form:
\text{ax + by = c}

Here, \(a, b, \text{and} c\) are constants. The most common form of a linear equation is the slope-intercept form:

\text{{y = mx + b}}

where \(m\) is the slope and \(b\) is the y-intercept (where the line crosses the y-axis). Understanding linear equations allows you to model situations like the access ramp: it translates real-world information, such as rise and run, into a mathematical relationship.

Slope

The slope is a crucial concept when dealing with linear equations. It represents the steepness and direction of a line. Mathematically, slope is defined as the ratio of the vertical change (\text{{rise}}) to the horizontal change (\text{{run}}) between two points on a line.

The formula for calculating the slope (\text{m}) is:\br> \text{{m = \(\frac{\text{rise}}{\text{run}}\)}}

In our ramp problem, the slope represents how much the ramp drops for each unit of horizontal distance. We calculated the slope as:
\(\text{{m = \)\frac{-2}{25}\(}}\)

The negative sign shows that the ramp is descending. Understanding slope helps you analyze and compare different lines. For instance, you can see if they are steep or gentle, or if they ascend or descend.

X-intercept

The x-intercept is where a line crosses the x-axis. At this point, the value of y is zero. To find the x-intercept, you set y to zero in the equation and solve for x.

In our ramp equation, \text{{y = -\(\frac{2}{25}\)x + 30}}, we set y to zero and solve for x:

0 = \text{-\(\frac{2}{25}\)x + 30} \text{x = 375}

This calculation shows that the ramp reaches the floor (when y = 0) 375 inches away from the starting platform. The x-intercept offers vital information about the specific point when another variable becomes zero. Understanding the x-intercept is significant for mapping where a particular change or event occurs in a graphical representation.

Design Requirements

Design requirements are crucial in ensuring that a structure is safe and functional. For our ramp, the design criteria include a maximum run of 30 feet and a maximum slope of a 1-inch drop for every 12 inches of run.

First, we convert the maximum run from feet to inches:
1 \text{foot} = 12 \text{inches}
\text{30 feet} = 30 x 12 = 360 \text{inches}
Since the needed run (375 inches) is longer than the maximum allowed (360 inches), the ramp does not meet the maximum run requirement.

Next, we check the slope requirement:
Maximum allowable slope = \(\frac{\text{1 inch drop}}{\text{12 inches run}}\)= \text{\(\frac{1}{12}\)}
Our slope (~0.08) is less than the maximum allowable slope (~0.083), meeting this requirement.

Design considerations help ensure safety and usability, illustrating why analyzing these factors critically is important.