Question

the manager of a restaurant found that the cost to produce 100 cups of coffee is $10.41, while the cost to produce 500 cups is $46.81. 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)=\boxed{} (use integers or decimals for any numbers in the expression.)
b. what is the fixed cost?
the fixed cost is $\boxed{}.
(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 $\boxed{}.
(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 $\boxed{}.
(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 \boxed{}\text{\textcent}.
(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 \boxed{}\text{\textcent}.
(type an integer or a decimal.)
what does the marginal cost of a cup of coffee mean to the manager?
\bigcirc a. the marginal cost of a cup of coffee is the cost of producing the first cup.
\bigcirc b. the marginal cost of a cup of coffee is the cost of producing one additional cup.
\bigcirc c. the marginal cost of a cup of coffee is the cost of producing a given number of cups.

Explanation:

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

A linear function has the form \( C(x) = mx + b \), where \( m \) is the slope (marginal cost) and \( b \) is the y-intercept (fixed cost). We have two points: \( (x_1, C(x_1)) = (100, 10.41) \) and \( (x_2, C(x_2)) = (500, 46.81) \).

Step 1: Calculate the slope \( m \)

The slope formula is \( m = \frac{C(x_2) - C(x_1)}{x_2 - x_1} \).
Substituting the values: \( m = \frac{46.81 - 10.41}{500 - 100} = \frac{36.4}{400} = 0.091 \).

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

Use the point \( (100, 10.41) \) and the slope \( m = 0.091 \) in \( C(x) = mx + b \).
\( 10.41 = 0.091(100) + b \)
\( 10.41 = 9.1 + b \)
Subtract 9.1 from both sides: \( b = 10.41 - 9.1 = 1.31 \).

So the formula is \( C(x) = 0.091x + 1.31 \).

Part b: Fixed cost

The fixed cost is the y-intercept \( b \) of the linear cost function. From part a, \( b = 1.31 \).

Part c: Total cost of producing 1200 cups

Substitute \( x = 1200 \) into \( C(x) = 0.091x + 1.31 \).

Step 1: Substitute \( x = 1200 \)

\( C(1200) = 0.091(1200) + 1.31 \)
\( C(1200) = 109.2 + 1.31 = 110.51 \).

Part d: Total cost of producing 1201 cups

Answer:

The marginal cost of a linear cost function is the slope \( m = 0.091 \) dollars, or 9.1 cents. The marginal cost of a cup of coffee is the cost of producing one additional cup (option B).

Final Answers:

a. \( C(x) = \boldsymbol{0.091x + 1.31} \)
b. The fixed cost is \( \boldsymbol{\$1.31} \)
c. The total cost of producing 1200 cups is \( \boldsymbol{\$110.51} \)
d. The total cost of producing 1201 cups is \( \boldsymbol{\$110.60} \)
e. The marginal cost of the 1201st cup is \( \boldsymbol{9.1} \) ¢
f. The marginal cost of any cup is \( \boldsymbol{9.1} \) ¢, and the correct option is \( \boldsymbol{B} \).