To solve this problem, we need to find the relationship between the weight of the barrel and the weight of the water it contains. We'll do this by setting up a system of equations based on the given information and solving for the variables involved.
### Given Information:
- The weight of the empty barrel is 20 pounds.
- When there are 10 gallons of water in the barrel, the total weight is 103.4 pounds.
- When there are 20 gallons of water in the barrel, the total weight is 186.8 pounds.
### Let’s define:
- [tex]\( y \)[/tex] as the total weight of the barrel and the water.
- [tex]\( x \)[/tex] as the gallons of water in the barrel.
- [tex]\( m \)[/tex] as the weight per gallon of water.
- [tex]\( c \)[/tex] as the weight of the empty barrel, which is 20 pounds.
### Step-by-Step Solution:
1. Set Up the System of Equations:
- For 10 gallons of water:
[tex]\[
y = mx + c
\][/tex]
[tex]\[
103.4 = 10m + 20
\][/tex]
- For 20 gallons of water:
[tex]\[
186.8 = 20m + 20
\][/tex]
2. Subtraction to Eliminate [tex]\( c \)[/tex]:
[tex]\[
(186.8 - 103.4) = (20m + 20) - (10m + 20)
\][/tex]
[tex]\[
83.4 = 10m
\][/tex]
3. Solve for [tex]\( m \)[/tex] (Weight per gallon of water):
[tex]\[
m = \frac{83.4}{10}
\][/tex]
[tex]\[
m = 8.34
\][/tex]
So, the weight of 1 gallon of water is [tex]\( 8.34 \)[/tex] pounds.
4. Write the Equation of the Line:
[tex]\( y = mx + c \)[/tex]
[tex]\[
y = 8.34x + 20
\][/tex]
### Final Answer:
The equation that matches this situation is:
[tex]\[
y = 8.34x + 20
\][/tex]
This linear equation represents the total weight ([tex]\( y \)[/tex]) of the barrel and water in terms of the gallons of water ([tex]\( x \)[/tex]).
### Graphing the Equation:
- The y-intercept [tex]\( c \)[/tex] is 20, indicating the weight of the empty barrel.
- The slope [tex]\( m \)[/tex] is 8.34, indicating the weight added per gallon of water.
As a result, the equation that matches the situation is indeed:
[tex]\[ \boxed{y = 8.34x + 20} \][/tex]
