Answered

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?


A$\frac{1}{3}$ inch up per foot across
B$\frac{1}{2}$ inch up per foot across
C2 inches up per foot across
D3 inches up per foot across

Verified Answer :

  • Identify the points provided as \((4, 12)\) and \((6, 18)\) in the coordinate system of (distance, height).
  • Apply the slope formula \(m = \frac{h_2 - h_1}{x_2 - x_1}\) to determine the rate of change.
  • Calculate the values: \(m = \frac{18 - 12}{6 - 4} = \frac{6}{2} = 3\).
  • Conclude that the ramp rises at a rate of 3 inches up per foot across, which is \(\boxed{3}\).

Explanation

  1. Problem Analysis

The problem describes a linear relationship between the height of a ramp \(h(x)\) in inches and the horizontal distance \(x\) from the driveway in feet. We are given two points on this line: \((x_1, h_1) = (4, 12)\) and \((x_2, h_2) = (6, 18)\). The rate of change of the incline corresponds to the slope \(m\) of this linear function.

  1. Calculating the Slope

For a linear function \(h(x) = mx + c\), the slope \(m\) represents the rate of change and is calculated using the formula:
\[m = \frac{h_2 - h_1}{x_2 - x_1}\]
Substituting the given coordinates into this formula:
\[m = \frac{18 \text{ inches} - 12 \text{ inches}}{6 \text{ feet} - 4 \text{ feet}}\]
\[m = \frac{6 \text{ inches}}{2 \text{ feet}}\]
\[m = 3 \text{ inches per foot}\]

  1. Final Conclusion

The calculated slope \(m = 3\) means that for every additional foot of horizontal distance, the ramp rises by \(3\) inches. Therefore, the rate of change of the ramp's incline is \(3\) inches up per foot across. This corresponds to the fourth option provided in the problem. The final answer is \(\boxed{3}\).

Examples

Understanding the slope of an incline is essential in construction and engineering. For instance, architects use these calculations to design wheelchair ramps that comply with accessibility standards, ensuring they are not too steep for safety. Similarly, civil engineers apply these principles when building roads and bridges to manage water drainage and ensure vehicle stability on sloped surfaces. By calculating the rate of change, you can ensure that any ramp or slope is safe, functional, and meets specific design requirements.

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