Question

the manager of a restaurant found that the cost to produce 300 cups of coffee is $29.55, while the cost to produce 500 cups is $48.35. 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)=0.094x + 1.35 (use integers or decimals for any numbers in the expression.)
b. what is the fixed cost?
the fixed cost is $1.35.
(type an integer or decimal rounded to two decimal places as needed.)
c. find the total cost of producing 1100 cups.
the total cost of producing 1100 cups is $104.75
(type an integer or decimal rounded to two decimal places as needed.)
d. find the total cost of producing 1101 cups.
the total cost of producing 1101 cups is $104.84
(type an integer or decimal rounded to two decimal places as needed.)
e. find the marginal cost of the 1101st cup.
the marginal cost of the 1101st cup is 9.4¢.
(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.)

Explanation:

Step1: Recall linear - function form

A linear function is of the form $C(x)=mx + b$, where $m$ is the slope and $b$ is the y - intercept (fixed cost). We are given two points $(x_1,C(x_1))=(300,29.55)$ and $(x_2,C(x_2))=(500,46.35)$.

Step2: Calculate the slope $m$

The slope $m=\frac{C(x_2)-C(x_1)}{x_2 - x_1}=\frac{46.35 - 29.55}{500 - 300}=\frac{16.8}{200}=0.084$.

Step3: Find the fixed - cost $b$

Substitute $m = 0.084$, $x = 300$ and $C(x)=29.55$ into $C(x)=mx + b$. Then $29.55=0.084\times300 + b$. Solving for $b$: $29.55 = 25.2+b$, so $b=29.55 - 25.2 = 4.35$. The cost function is $C(x)=0.084x + 4.35$.

Step4: Calculate total cost for 1100 cups

Substitute $x = 1100$ into $C(x)=0.084x + 4.35$. Then $C(1100)=0.084\times1100+4.35=92.4 + 4.35=96.75$.

Step5: Calculate total cost for 1101 cups

Substitute $x = 1101$ into $C(x)=0.084x + 4.35$. Then $C(1101)=0.084\times1101+4.35=(0.084\times1100+0.084)+4.35=92.4+0.084 + 4.35=96.834\approx96.83$.

Step6: Calculate marginal cost of 1101st cup

The marginal cost is the change in cost when one more unit is produced. $MC = C(1101)-C(1100)=96.83 - 96.75 = 0.084$ or 8.4 cents.

Step7: Marginal cost of any cup

For a linear cost function $C(x)=mx + b$, the marginal cost is the slope of the cost - function. So the marginal cost of any cup is $0.084$ or 8.4 cents. This means that for each additional cup of coffee produced, the cost to the manager increases by 8.4 cents.

Answer:

a. $C(x)=0.084x + 4.35$
b. $4.35$
c. $96.75$
d. $96.83$
e. $8.4$
f. $8.4$