College

Assuming the total cost is a linear function, write an equation for the cost of producing [tex]x[/tex] bar stools.

Given:
- The cost of producing 370 bar stools is [tex]$3125[/tex].
- The cost of producing 750 bar stools is [tex]$5785[/tex].

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.