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Let's go through the question one step at a time.

### (a) What is the maximum allowable slope for a wheelchair ramp?

The problem states that the ramp must have a maximum rise of 1 inch for every 12 inches of horizontal distance. The slope \( m \) of a ramp is the ratio of the vertical rise to the horizontal run. Therefore, the maximum slope is:
\[
m = \frac{1 \, \text{inch}}{12 \, \text{inches}} = \frac{1}{12}
\]

So, the maximum allowable slope is \( \frac{1}{12} \).

### Now, to find the function \( H(x) \):

The linear function \( H(x) \) that models the height of the ramp above the ground can be written in the form:
\[
H(x) = m \cdot x
\]
where \( m = \frac{1}{12} \) is the slope and \( x \) is the horizontal distance. Therefore, the function is:
\[
H(x) = \frac{1}{12} x
\]

### (b) If the space available to build a ramp is 162 inches wide, how high (in inches) does the ramp reach?

To calculate the height, we use the linear function found above, substituting \( x = 162 \) inches into the equation:
\[
H(162) = \frac{1}{12} \times 162 = 13.5 \, \text{inches}
\]

Thus, the ramp reaches a height of 13.5 inches.

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Would you like further details or explanations on any of these parts?

Here are five related questions you might find interesting:
1. How would the function \( H(x) \) change if the slope requirement was steeper, say 1 inch for every 10 inches?
2. What is the importance of ensuring a maximum slope for accessibility in public spaces?
3. How long would a ramp need to be to reach a height of 24 inches?
4. Could the slope of the ramp be different for indoor versus outdoor settings?
5. How is the slope affected by using metric units instead of inches?

**Tip:** When working with slope problems, remember that the units should always be consistent (e.g., both in inches or both in meters) to avoid mistakes in the calculation.

A local diner must build a wheelchair ramp to provide accessibility. Federal building codes require a maximum rise of 1 inch for every 12 inches of horizontal distance. (a) What is the maximum allowable slope for a wheelchair ramp? (b) If the space available to build a ramp is 162 inches wide, how high does the ramp reach?

Learn how to calculate the maximum allowable slope for a wheelchair ramp and determine the ramp's height based on available space. This problem covers key concepts like slope and linear equations, suitable for students in Grades 7-9.

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