A school system is reducing the amount of dumpster loads of trash removed each week. In week 5, there were 60 dumpster loads of waste removed. In week 10, there were 40 dumpster loads removed. Assume that the reduction in the amount of waste each week is linear. Write an equation in function form to show the amount of trash removed each week.

To find the linear equation that models the amount of trash removed each week, we will treat this as a linear relationship between the week number and the amount of dumpster loads of trash removed.

The linear equation will have the general form:

\[
y = mx + b
\]

Where:
- \(y\) is the number of dumpster loads removed in a given week,
- \(x\) is the week number,
- \(m\) is the slope (rate of reduction per week),
- \(b\) is the y-intercept (the initial amount of trash removed at week 0).

### Step 1: Calculate the slope \(m\)
We are given two points:
- In week 5, 60 dumpster loads were removed: \( (x_1, y_1) = (5, 60) \),
- In week 10, 40 dumpster loads were removed: \( (x_2, y_2) = (10, 40) \).

The slope \(m\) is calculated as:

\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{40 - 60}{10 - 5} = \frac{-20}{5} = -4
\]

This means the number of dumpster loads is decreasing by 4 loads per week.

### Step 2: Find the y-intercept \(b\)
Using the point-slope form, we can now substitute the slope \(m = -4\) and one of the given points, say \( (5, 60) \), into the equation \( y = mx + b \) to solve for \(b\):

\[
60 = (-4)(5) + b
\]
\[
60 = -20 + b
\]
\[
b = 80
\]

### Step 3: Write the equation
Now that we know the slope \(m = -4\) and the y-intercept \(b = 80\), we can write the equation for the amount of trash removed each week:

\[
y = -4x + 80
\]

### Conclusion:
The equation \(y = -4x + 80\) shows the amount of dumpster loads of trash removed in week \(x\). Each week, the trash amount decreases by 4 loads, starting from 80 loads in week 0.

---

Would you like any details or have questions? Here are some related questions for further exploration:

1. How many dumpster loads will be removed in week 15?
2. In what week will the amount of trash removed reach zero?
3. What does the slope of \(-4\) represent in the context of this problem?
4. How much trash was likely removed in the first week (week 1)?
5. What would the equation look like if the amount of trash was reduced by 3 loads per week instead?

**Tip:** When solving linear equations, interpreting the slope as the rate of change helps to understand how values evolve over time.

Learn how to model a real-world problem involving trash reduction using linear equations. In this example, a school system reduces trash by 4 dumpster loads per week, starting with 80 loads in week 0. This solution walks through the slope-intercept form and explains the process for solving. Suitable for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation