Answer :
To write an equation for the cost of producing x bar stools, given that the cost is a linear function, we need to determine the relationship between the number of stools produced and the total cost.
We are given two data points:
1. Producing 370 bar stools costs [tex]$3125.
2. Producing 750 bar stools costs $[/tex]5785.
Since the cost is assumed to be a linear function, it follows the format [tex]\( C(x) = mx + b \)[/tex], where:
- [tex]\( C(x) \)[/tex] is the total cost for producing [tex]\( x \)[/tex] stools.
- [tex]\( m \)[/tex] is the slope of the line, representing the cost per stool.
- [tex]\( b \)[/tex] is the y-intercept, representing the fixed costs (cost when no stools are produced).
Step 1: Calculate the slope ([tex]\( m \)[/tex])
The slope [tex]\( m \)[/tex] can be calculated using the formula for the slope of a line between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex]:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
For our points (370, 3125) and (750, 5785):
- [tex]\( x_1 = 370, y_1 = 3125 \)[/tex]
- [tex]\( x_2 = 750, y_2 = 5785 \)[/tex]
[tex]\[
m = \frac{5785 - 3125}{750 - 370} = \frac{2660}{380} = 7.0
\][/tex]
So, the cost per stool is [tex]$7.00.
Step 2: Calculate the y-intercept (\( b \))
We can use one of the data points and the slope to solve for \( b \) using the equation of the line:
\[
y = mx + b
\]
Substitute \( x = 370, y = 3125, \) and \( m = 7.0 \):
\[
3125 = 7.0 \times 370 + b
\]
\[
3125 = 2590 + b
\]
\[
b = 3125 - 2590 = 535
\]
So, the fixed cost is $[/tex]535.
Final Equation
Putting it all together, the equation for the cost as a function of [tex]\( x \)[/tex], the number of stools, is:
[tex]\[
C(x) = 7.0 \times x + 535
\][/tex]
This equation represents the linear relationship between the number of bar stools produced and the total cost.
We are given two data points:
1. Producing 370 bar stools costs [tex]$3125.
2. Producing 750 bar stools costs $[/tex]5785.
Since the cost is assumed to be a linear function, it follows the format [tex]\( C(x) = mx + b \)[/tex], where:
- [tex]\( C(x) \)[/tex] is the total cost for producing [tex]\( x \)[/tex] stools.
- [tex]\( m \)[/tex] is the slope of the line, representing the cost per stool.
- [tex]\( b \)[/tex] is the y-intercept, representing the fixed costs (cost when no stools are produced).
Step 1: Calculate the slope ([tex]\( m \)[/tex])
The slope [tex]\( m \)[/tex] can be calculated using the formula for the slope of a line between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex]:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
For our points (370, 3125) and (750, 5785):
- [tex]\( x_1 = 370, y_1 = 3125 \)[/tex]
- [tex]\( x_2 = 750, y_2 = 5785 \)[/tex]
[tex]\[
m = \frac{5785 - 3125}{750 - 370} = \frac{2660}{380} = 7.0
\][/tex]
So, the cost per stool is [tex]$7.00.
Step 2: Calculate the y-intercept (\( b \))
We can use one of the data points and the slope to solve for \( b \) using the equation of the line:
\[
y = mx + b
\]
Substitute \( x = 370, y = 3125, \) and \( m = 7.0 \):
\[
3125 = 7.0 \times 370 + b
\]
\[
3125 = 2590 + b
\]
\[
b = 3125 - 2590 = 535
\]
So, the fixed cost is $[/tex]535.
Final Equation
Putting it all together, the equation for the cost as a function of [tex]\( x \)[/tex], the number of stools, is:
[tex]\[
C(x) = 7.0 \times x + 535
\][/tex]
This equation represents the linear relationship between the number of bar stools produced and the total cost.
