Question

the manager of a restaurant found that the cost to produce 300 cups of coffee is $30.44, while the cost to produce 400 cups is $40.14. assume the cost c(x) is a linear function of x, the number of cups produced. answer parts a through f.
a. find a formula for c(x). choose the correct answer below.
c(x)=□ (use integers or decimals for any numbers in the expression.)
b. what is the fixed cost?
the fixed cost is $□
(type an integer or decimal rounded to two decimal places as needed.)
c. find the total cost of producing 1200 cups.
the total cost of producing 1200 cups is $□
(type an integer or decimal rounded to two decimal places as needed.)
d. find the total cost of producing 1201 cups.
the total cost of producing 1201 cups is $□
(type an integer or decimal rounded to two decimal places as needed.)
e. find the marginal cost of the 1201st cup.
the marginal cost of the 1201st cup is □¢
(type an integer or a decimal.)
f. what is the marginal cost of any cup and what does this mean to the manager?
the marginal cost of any cup is □¢
(type an integer or a decimal.)
what does the marginal cost of a cup of coffee mean to the manager?
○ a. the marginal cost of a cup of coffee is the cost of producing the first cup.
○ b. the marginal cost of a cup of coffee is the cost of producing a given number of cups.
○ c. the marginal cost of a cup of coffee is the cost of producing one additional cup.

Explanation:

Part a: Find a formula for \( C(x) \)

Step 1: Find the slope \( m \)

The slope \( m \) of a linear function \( C(x) = mx + b \) is calculated using two points \( (x_1, C(x_1)) \) and \( (x_2, C(x_2)) \). Here, \( x_1 = 300 \), \( C(x_1) = 30.44 \), \( x_2 = 400 \), \( C(x_2) = 40.14 \). The formula for slope is \( m=\frac{C(x_2)-C(x_1)}{x_2 - x_1} \).
\[
m=\frac{40.14 - 30.44}{400 - 300}=\frac{9.7}{100}=0.097
\]

Step 2: Find the y - intercept \( b \)

Use the point - slope form \( C(x)-C(x_1)=m(x - x_1) \) or substitute \( x = 300 \), \( C(x)=30.44 \) and \( m = 0.097 \) into \( C(x)=mx + b \).
\[
30.44=0.097\times300 + b
\]
\[
30.44 = 29.1+b
\]
Subtract 29.1 from both sides: \( b=30.44 - 29.1 = 1.34 \)
So the formula for \( C(x) \) is \( C(x)=0.097x + 1.34 \)

The fixed cost is the value of \( C(x) \) when \( x = 0 \) (the y - intercept of the linear cost function \( C(x)=mx + b \)). From the formula \( C(x)=0.097x + 1.34 \), when \( x = 0 \), \( C(0)=1.34 \)

Substitute \( x = 1200 \) into the cost function \( C(x)=0.097x+1.34 \)
\[
C(1200)=0.097\times1200 + 1.34
\]
\[
C(1200)=116.4+1.34=117.74
\]

Answer:

\( C(x)=0.097x + 1.34 \)

Part b: What is the fixed cost?