Let's solve the problem step by step:
Part (a): Writing the Linear Equation
Given points:
- \((x_1, y_1) = (50, 27.50)\)
- \((x_2, y_2) = (400, 115.00)\)
A linear equation can be written in the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
1. Calculate the slope \(m\):
[tex]\[
m = \frac{(y_2 - y_1)}{(x_2 - x_1)}
\][/tex]
[tex]\[
m = \frac{(115.00 - 27.50)}{(400 - 50)} = \frac{87.50}{350} = 0.25
\][/tex]
2. Calculate the y-intercept \(b\):
Using the point \((x_1, y_1)\):
[tex]\[
b = y_1 - mx_1
\][/tex]
[tex]\[
b = 27.50 - (0.25 \times 50) = 27.50 - 12.50 = 15.00
\][/tex]
So, the linear equation is:
[tex]\[
y = 0.25x + 15.0
\][/tex]
Part (b): Finding the Number of Cups Produced for $130.00
Given the linear equation we derived:
[tex]\[
y = 0.25x + 15.0
\][/tex]
We need to find \(x\) when the cost \(y\) is $130.00:
[tex]\[
130.00 = 0.25x + 15.0
\][/tex]
Solving for \(x\):
[tex]\[
130.00 - 15.00 = 0.25x
\][/tex]
[tex]\[
115.00 = 0.25x
\][/tex]
[tex]\[
x = \frac{115.00}{0.25} = 460
\][/tex]
Summary:
- The linear equation that expresses the cost \(y\) in terms of the number of cups of coffee \(x\) is \(y = 0.25x + 15.0\).
- If the cost of production is $130.00, the total number of cups produced is [tex]\(460\)[/tex] cups.
