To determine the range of the function that models the height of the ramp $y$ in feet, given the horizontal distance $x$ from the top of the ramp, we need to model the situation first.

Understanding the Problem:

• The ramp's height starts at 8 feet when $x = 0$ (the top of the ramp).
• The height decreases linearly to 0 feet at $x = 20$ feet (the bottom of the ramp).

This suggests that the height of the ramp $y$ as a function of the horizontal distance $x$ is a linear function, decreasing from 8 feet to 0 feet as $x$ increases from 0 to 20 feet.

Linear Function:

Since the height decreases linearly, we can represent this situation with a linear equation of the form: $y = mx + b$ Where:

• $m$ is the slope (rate of change of height with respect to $x$),
• $b$ is the y-intercept (the height when $x = 0$).

Given that the height at the top of the ramp ($x = 0$) is 8 feet, the y-intercept $b = 8$.

Now, we need to find the slope $m$. We know that at $x = 20$, the height is 0 feet, so we can use the two points $(0, 8)$ and $(20, 0)$ to calculate the slope:

$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - 8}{20 - 0} = \frac{-8}{20} = -\frac{2}{5}$

Thus, the equation for the height of the ramp as a function of horizontal distance is:

$y = -\frac{2}{5}x + 8$

Range of the Function:

The range refers to the possible values of $y$ (the height of the ramp) as $x$ varies from 0 to 20.

• When $x = 0$, $y = 8$ (the top of the ramp).
• When $x = 20$, $y = 0$ (the bottom of the ramp).

Since the height decreases linearly from 8 feet to 0 feet as $x$ increases from 0 to 20, the height $y$ will take all values between 0 and 8. Therefore, the range of the function is:

$\boxed{[0, 8]}$

Summary:

The range of the function that models the height of the ramp is the set of values from 0 feet to 8 feet, inclusive.

Would you like more details or have any other questions?

Here are some related questions:

1. How do you interpret the slope of a linear function in real-world contexts?
2. How would the function change if the ramp had a steeper slope?
3. Can you graph the linear function $y = -\frac{2}{5}x + 8$?
4. How do you find the domain of a linear function in a real-world scenario?
5. What are some other examples of linear relationships in everyday life?

Tip: For any linear function, you can always use two points to calculate the slope and write the equation of the line.