https://file.aiquickdraw.com/mathbot/file/17299027006475qaxecf4.jpg

Let's solve the problem step-by-step:

1. **Identify the Points:** We have two data points based on the driver's distance from Reno after certain times:
   - After 1 hour, the driver is 84 miles from Reno. So, we have the point \((1, 84)\).
   - After 2 hours, the driver is 36 miles from Reno. So, we have the point \((2, 36)\).

2. **Calculate the Slope (m):** The slope of a line, \( m \), is calculated by:
   \[
   m = \frac{{y_2 - y_1}}{{x_2 - x_1}}
   \]
   Plugging in the points:
   \[
   m = \frac{{36 - 84}}{{2 - 1}} = \frac{{-48}}{{1}} = -48
   \]
   So, the slope \( m = -48 \).

3. **Write the Equation in Point-Slope Form:** Using the point-slope form \( y - y_1 = m(x - x_1) \) with one of the points, say \((1, 84)\):
   \[
   y - 84 = -48(x - 1)
   \]
   Simplifying, we get:
   \[
   y = -48x + 132
   \]

   Therefore, the linear function that relates the time \( x \) in hours since leaving Sacramento to the driver’s distance \( y \) in miles from Reno is:
   \[
   y = -48x + 132
   \]

4. **Interpret the Slope and y-Intercept:**
   - **Slope (-48):** This represents the rate at which the driver is approaching Reno. Specifically, the driver is getting 48 miles closer to Reno per hour.
   - **y-Intercept (132):** This represents the initial distance from Reno when \( x = 0 \), meaning the driver started 132 miles away from Reno.

Would you like more details or have any questions?

---

Here are some related questions to deepen your understanding:

1. What would the equation be if the driver had started 150 miles away from Reno?
2. How would the slope change if the driver were moving at a slower speed?
3. What does it mean if the slope of this function were positive?
4. How would the equation change if the driver’s starting distance was unknown?
5. If the driver traveled at a constant speed for 3 hours, what would their distance from Reno be?

**Tip:** Remember, the y-intercept in a distance-time relationship often represents the starting point in terms of distance from a destination.

A truck driver is traveling from Sacramento to Reno. After 1 hour, the driver is 84 miles from Reno. After 2 hours, the driver is 36 miles from Reno. Write the linear function that relates the time x in hours since leaving Sacramento to the driver’s distance y in miles from Reno. Explain the meaning of the slope and y-intercept in this situation.

Learn how to write a linear function relating time and distance for a truck traveling from Sacramento to Reno. This problem explores the slope as a rate of approach to Reno and the y-intercept as the initial distance, providing a detailed step-by-step solution. Ideal for Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation