Answer :
To find the linear equation for the cost of a gallon of milk, [tex]\( C \)[/tex], in terms of the number of years since 1980, [tex]\( t \)[/tex], we'll follow these steps:
1. Identify the data points: We know that in 1985, the cost was \[tex]$2.20, and in 2023, the cost was \$[/tex]4.10. Since [tex]\( t \)[/tex] is the number of years since 1980, for 1985, [tex]\( t = 5 \)[/tex] (because 1985 - 1980 = 5) and for 2023, [tex]\( t = 43 \)[/tex] (because 2023 - 1980 = 43).
2. Calculate the slope of the line: The slope of a linear equation is calculated as the change in cost divided by the change in time. Here, it’s:
[tex]\[
\text{slope} = \frac{\text{cost in 2023} - \text{cost in 1985}}{\text{year 2023} - \text{year 1985}} = \frac{4.10 - 2.20}{43 - 5} = \frac{1.90}{38} \approx 0.05
\][/tex]
3. Find the y-intercept: The equation of the line is generally written as [tex]\( C = mt + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept. We already determined that the slope [tex]\( m \approx 0.05 \)[/tex]. Using the point for 1985 [tex]\((t = 5, C = 2.20)\)[/tex] to find [tex]\( b \)[/tex]:
[tex]\[
2.20 = 0.05 \times 5 + b \\
2.20 = 0.25 + b \\
b = 2.20 - 0.25 = 1.95
\][/tex]
4. Write out the equation: Now we can write the linear equation for the cost in terms of [tex]\( t \)[/tex]:
[tex]\[
C = 0.05t + 1.95
\][/tex]
This is the linear equation describing the relationship between the number of years since 1980, [tex]\( t \)[/tex], and the cost of a gallon of milk, [tex]\( C \)[/tex]. So, the correct answer is [tex]\( C = 0.05t + 1.95 \)[/tex].
1. Identify the data points: We know that in 1985, the cost was \[tex]$2.20, and in 2023, the cost was \$[/tex]4.10. Since [tex]\( t \)[/tex] is the number of years since 1980, for 1985, [tex]\( t = 5 \)[/tex] (because 1985 - 1980 = 5) and for 2023, [tex]\( t = 43 \)[/tex] (because 2023 - 1980 = 43).
2. Calculate the slope of the line: The slope of a linear equation is calculated as the change in cost divided by the change in time. Here, it’s:
[tex]\[
\text{slope} = \frac{\text{cost in 2023} - \text{cost in 1985}}{\text{year 2023} - \text{year 1985}} = \frac{4.10 - 2.20}{43 - 5} = \frac{1.90}{38} \approx 0.05
\][/tex]
3. Find the y-intercept: The equation of the line is generally written as [tex]\( C = mt + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept. We already determined that the slope [tex]\( m \approx 0.05 \)[/tex]. Using the point for 1985 [tex]\((t = 5, C = 2.20)\)[/tex] to find [tex]\( b \)[/tex]:
[tex]\[
2.20 = 0.05 \times 5 + b \\
2.20 = 0.25 + b \\
b = 2.20 - 0.25 = 1.95
\][/tex]
4. Write out the equation: Now we can write the linear equation for the cost in terms of [tex]\( t \)[/tex]:
[tex]\[
C = 0.05t + 1.95
\][/tex]
This is the linear equation describing the relationship between the number of years since 1980, [tex]\( t \)[/tex], and the cost of a gallon of milk, [tex]\( C \)[/tex]. So, the correct answer is [tex]\( C = 0.05t + 1.95 \)[/tex].