To solve this problem, we need to determine a linear function that models the relationship between the tank size (in gallons) and the total cost for a fill-up and a car wash. We can use the slope-intercept form of a linear equation, which is represented as [tex]\( f(x) = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
The data provided gives us the following points:
- (9, 19.85)
- (14, 29.35)
- (20, 40.75)
### Step-by-Step Solution:
1. Calculate the slope (m):
The slope [tex]\( m \)[/tex] is calculated using the formula:
[tex]\[
m = \frac{n\sum{xy} - (\sum{x})(\sum{y})}{n\sum{x^2} - (\sum{x})^2}
\][/tex]
Where:
- [tex]\( n \)[/tex] is the number of data points.
- [tex]\( \sum{xy} \)[/tex] is the sum of the products of the tank sizes and their corresponding costs.
- [tex]\( \sum{x} \)[/tex] is the sum of the tank sizes.
- [tex]\( \sum{y} \)[/tex] is the sum of the costs.
- [tex]\( \sum{x^2} \)[/tex] is the sum of the squares of the tank sizes.
Given values:
[tex]\[
n = 3
\][/tex]
[tex]\[
\sum{x} = 9 + 14 + 20 = 43
\][/tex]
[tex]\[
\sum{y} = 19.85 + 29.35 + 40.75 = 89.95
\][/tex]
[tex]\[
\sum{x^2} = 9^2 + 14^2 + 20^2 = 81 + 196 + 400 = 677
\][/tex]
[tex]\[
\sum{xy} = (9 \times 19.85) + (14 \times 29.35) + (20 \times 40.75) = 178.65 + 410.9 + 815 = 1404.55
\][/tex]
Plugging these into the slope formula:
[tex]\[
m = \frac{3 \times 1404.55 - 43 \times 89.95}{3 \times 677 - 43^2}
\][/tex]
[tex]\[
m = \frac{4213.65 - 3867.85}{2031 - 1849}
\][/tex]
[tex]\[
m = \frac{345.8}{182} \approx 1.90
\][/tex]
2. Calculate the intercept (b):
The y-intercept [tex]\( b \)[/tex] is calculated using the formula:
[tex]\[
b = \frac{\sum{y} - m \sum{x}}{n}
\][/tex]
[tex]\[
b = \frac{89.95 - 1.90 \times 43}{3}
\][/tex]
[tex]\[
b = \frac{89.95 - 81.7}{3}
\][/tex]
[tex]\[
b \approx \frac{8.25}{3} = 2.75
\][/tex]
So, the linear equation is:
[tex]\[
f(x) = 1.90x + 2.75
\][/tex]
3. Find the cost for a truck with a tank size of 31 gallons:
Substitute [tex]\( x = 31 \)[/tex] into the equation:
[tex]\[
f(31) = 1.90 \times 31 + 2.75
\][/tex]
[tex]\[
f(31) = 58.9 + 2.75
\][/tex]
[tex]\[
f(31) = 61.65
\][/tex]
Therefore, the equation for the function is [tex]\( f(x) = 1.90x + 2.75 \)[/tex], and the cost for a fill-up and a car wash for a customer with a truck whose tank size is 31 gallons is [tex]$61.65.
The correct answer is:
a. \( f(x) = 1.90x + 2.75 \); Cost for truck \( = \$[/tex]61.65 \)
