Problem 55
Question
The slope for a wheelchair ramp for a home has to be \(\frac{1}{12}\). If the vertical distance from the ground to the door bottom is \(2.5 \mathrm{ft}\), find the distance the ramp has to extend from the home in order to comply with the needed slope.
Step-by-Step Solution
Verified Answer
The ramp must extend 30 feet from the home.
1Step 1: Understand the Slope
The slope of a ramp is defined as the ratio of the vertical rise to the horizontal run. In this problem, the slope is given as \( \frac{1}{12} \), meaning for every 1 foot of vertical rise, the ramp must extend 12 feet horizontally.
2Step 2: Set up the Equation
The given vertical distance (rise) from the ground to the door is 2.5 feet. Using the formula \( \text{slope} = \frac{\text{rise}}{\text{run}} \), we can write the equation as \( \frac{1}{12} = \frac{2.5}{x} \), where \( x \) is the horizontal distance (run) the ramp must cover.
3Step 3: Solve for the Run
To find \( x \), cross-multiply the equation from Step 2: \( 1 \times x = 12 \times 2.5 \). Simplify this to \( x = 30 \).
4Step 4: Write the Conclusion
The ramp has to extend 30 feet from the home to have the required slope of \( \frac{1}{12} \).
Key Concepts
SlopeRatioHorizontal DistanceVertical Distance
Slope
The concept of slope is essential when analyzing ramps, hills, or any inclined surface. Slope basically describes how steep or gentle a surface is. In mathematical terms, slope is defined as the ratio of the vertical change (rise) to the horizontal change (run). So, when you hear someone mention the slope is \( \frac{1}{12} \), it means for every unit the surface rises vertically, it stretches out 12 units horizontally.
Understanding slope helps us comply with regulations, such as those for building ramps, which often specify a maximum steepness to ensure safety and accessibility. In our exercise, the specific slope ratio provided is crucial to calculate how long the ramp must extend horizontally, given a certain vertical rise. The ramp slope ensures comfort and safety for those using it and must be measured carefully.
Understanding slope helps us comply with regulations, such as those for building ramps, which often specify a maximum steepness to ensure safety and accessibility. In our exercise, the specific slope ratio provided is crucial to calculate how long the ramp must extend horizontally, given a certain vertical rise. The ramp slope ensures comfort and safety for those using it and must be measured carefully.
Ratio
A ratio is a way of comparing two quantities. It's an expression that shows how many times one quantity is contained within another. In the case of a wheelchair ramp's slope, the ratio \( \frac{1}{12} \) illustrates how the vertical rise is much smaller compared to the horizontal run.
The ratio allows anyone to scale the design of a ramp appropriately, regardless of how much vertical distance you need to cover. By ensuring that the ramp's slope sticks to the ratio of \( \frac{1}{12} \), architects and builders can create safe and accessible ramps for people in wheelchairs or with mobility issues. Ratios provide a simple yet powerful tool to maintain these consistent relationships as seen in practical applications like ramp construction.
The ratio allows anyone to scale the design of a ramp appropriately, regardless of how much vertical distance you need to cover. By ensuring that the ramp's slope sticks to the ratio of \( \frac{1}{12} \), architects and builders can create safe and accessible ramps for people in wheelchairs or with mobility issues. Ratios provide a simple yet powerful tool to maintain these consistent relationships as seen in practical applications like ramp construction.
Horizontal Distance
Horizontal distance refers to the length or extent of the ground covered from start to end along the horizontal plane. When constructing a ramp, this is often referred to as the 'run' and is critical in determining how gentle or steep the slope will be.
In our exercise, the horizontal distance is what we solve for once the vertical distance and slope ratio are known. We use the equation \( \frac{1}{12} = \frac{2.5}{x} \) to find the required horizontal distance. By calculating the horizontal distance of 30 feet, we've determined how far the ramp needs to extend outward from the house, ensuring compliance with the safe slope ratio. This horizontal spread provides ease of use without excessive exertion for users.
In our exercise, the horizontal distance is what we solve for once the vertical distance and slope ratio are known. We use the equation \( \frac{1}{12} = \frac{2.5}{x} \) to find the required horizontal distance. By calculating the horizontal distance of 30 feet, we've determined how far the ramp needs to extend outward from the house, ensuring compliance with the safe slope ratio. This horizontal spread provides ease of use without excessive exertion for users.
Vertical Distance
Vertical distance is simply the height difference between two points on a surface. In many practical situations, vertical distance is referred to as the 'rise' in the context of slopes and ramps.
For the problem at hand, the vertical distance is 2.5 feet — the height from the ground up to the bottom of the door. Knowing this vertical measurement is vital in setting up equations to solve for the necessary horizontal distance with the given slope of \( \frac{1}{12} \).
Correctly measuring vertical distance ensures that the ramp meets accessibility standards, leading to a safe and efficiently designed structure that can accommodate all users effectively.
For the problem at hand, the vertical distance is 2.5 feet — the height from the ground up to the bottom of the door. Knowing this vertical measurement is vital in setting up equations to solve for the necessary horizontal distance with the given slope of \( \frac{1}{12} \).
Correctly measuring vertical distance ensures that the ramp meets accessibility standards, leading to a safe and efficiently designed structure that can accommodate all users effectively.
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