Cars produce carbon dioxide pollution. The rate they produce the pollution is measured g/km (grams per kilometer). Suppose that a family's two cars produce a combined rate of 60 g/km of carbon dioxide pollution. During one day last week, the first car drove 35 kilometers and the second drove 20 kilometers. The two cars produced a combined total of 1800 grams of carbon dioxide pollution. At what rates does each car produce carbon dioxide pollution?

Let \( r_1 \) be the rate of carbon dioxide production for the first car in grams per kilometer (g/km), and let \( r_2 \) be the rate for the second car in g/km. We are given two pieces of information:

1. The combined rate of pollution for both cars is 60 g/km:
   \[
   r_1 + r_2 = 60
   \]

2. The combined total of carbon dioxide pollution produced in one day is 1800 grams. The first car drove 35 kilometers, and the second car drove 20 kilometers:
   \[
   35r_1 + 20r_2 = 1800
   \]

We now have a system of linear equations:
\[
\begin{cases}
r_1 + r_2 = 60 \\
35r_1 + 20r_2 = 1800
\end{cases}
\]

To solve this system, we can use substitution or elimination. We will use substitution. First, solve the first equation for \( r_1 \):
\[
r_1 = 60 - r_2
\]

Next, substitute \( r_1 \) into the second equation:
\[
35(60 - r_2) + 20r_2 = 1800
\]

Expand and simplify:
\[
2100 - 35r_2 + 20r_2 = 1800
\]
\[
2100 - 15r_2 = 1800
\]

Isolate \( r_2 \):
\[
2100 - 1800 = 15r_2
\]
\[
300 = 15r_2
\]
\[
r_2 = 20
\]

Substitute \( r_2 \) back into the first equation to find \( r_1 \):
\[
r_1 + 20 = 60
\]
\[
r_1 = 40
\]

Therefore, the rates at which each car produces carbon dioxide pollution are:
\[
r_1 = 40 \, \text{g/km}
\]
\[
r_2 = 20 \, \text{g/km}
\]

To verify, we can check the calculations:
- Combined rate: \( 40 + 20 = 60 \, \text{g/km} \)
- Total pollution produced:
  \[
  35 \times 40 + 20 \times 20 = 1400 + 400 = 1800 \, \text{grams}
  \]

Both conditions are satisfied. Hence, the solution is correct.

Would you like more details or have any questions?

Here are some related questions to expand your understanding:
1. What would the rates be if the total pollution produced was different?
2. How would the solution change if one of the cars had driven a different distance?
3. What would happen if the combined rate was different but the total pollution remained the same?
4. How can you solve the system using the elimination method?
5. What if there were three cars instead of two?
6. How does the method of substitution compare to the method of elimination?
7. What are the implications of higher pollution rates for car emissions?
8. How can these calculations be useful in real-world environmental policies?

**Tip:** When solving systems of equations, clearly label your variables and ensure all given conditions are incorporated into your equations for accurate solutions.

Learn how to calculate the rates at which two cars produce carbon dioxide pollution based on given distances and total pollution produced. This problem involves solving a system of linear equations and understanding pollution rates in grams per kilometer (g/km). Suitable for students in Grades 7-9.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation