To determine the rate of carbon dioxide production for each car, we can set up a system of equations based on the given information. We'll define the rates of carbon dioxide production for the first and second cars as [tex]\( r_1 \)[/tex] and [tex]\( r_2 \)[/tex] respectively.
Here's the information given:
- The combined rate of carbon dioxide production is [tex]\( 65 \frac{ \text{g} }{ \text{km} } \)[/tex].
- The first car drove 20 kilometers.
- The second car drove 40 kilometers.
- The combined total of carbon dioxide produced is 1800 grams.
From this information, we can derive the following equations:
1. The combined rate equation:
[tex]\[ r_1 + r_2 = 65 \][/tex]
2. The total carbon dioxide production equation (using the distance each car traveled):
[tex]\[ 20r_1 + 40r_2 = 1800 \][/tex]
We can solve these two equations step-by-step to find [tex]\( r_1 \)[/tex] and [tex]\( r_2 \)[/tex].
Step 1: Solve the combined rate equation for [tex]\( r_1 \)[/tex]:
[tex]\[ r_1 = 65 - r_2 \][/tex]
Step 2: Substitute this expression for [tex]\( r_1 \)[/tex] into the total carbon dioxide production equation:
[tex]\[ 20(65 - r_2) + 40r_2 = 1800 \][/tex]
Step 3: Distribute and combine like terms:
[tex]\[ 1300 - 20r_2 + 40r_2 = 1800 \][/tex]
[tex]\[ 1300 + 20r_2 = 1800 \][/tex]
Step 4: Isolate [tex]\( r_2 \)[/tex]:
[tex]\[ 20r_2 = 500 \][/tex]
[tex]\[ r_2 = 25 \][/tex]
Step 5: Substitute [tex]\( r_2 \)[/tex] back into the equation [tex]\( r_1 = 65 - r_2 \)[/tex]:
[tex]\[ r_1 = 65 - 25 \][/tex]
[tex]\[ r_1 = 40 \][/tex]
Thus, the rates of carbon dioxide production are:
- First car: [tex]\( 40 \frac{ \text{g} }{ \text{km} } \)[/tex]
- Second car: [tex]\( 25 \frac{ \text{g} }{ \text{km} } \)[/tex]
