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The manager of a restaurant found that the cost to produce 50 cups of coffee is [tex]\[tex]$27.50[/tex], while the cost to produce 400 cups is [tex]\$[/tex]115.00[/tex]. Assume the relationship between the cost [tex]y[/tex] to produce [tex]x[/tex] cups of coffee is linear.

a. Write a linear equation that expresses the cost, [tex]y[/tex], in terms of the number of cups of coffee, [tex]x[/tex].
b. How many cups of coffee are produced if the cost of production is [tex]\$130.00[/tex]?

a. [tex]y = 0.25x + 15[/tex] (Use integers or decimals for any numbers in the expression.)
b. The total number of cups produced for a cost of [tex]\$130.00[/tex] is [tex]\boxed{\text{ }}[/tex] cups. (Simplify your answer.)



Answer :

GinnyAnswer
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.

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