## Related questions with answers

The manager of a restaurant found that the cost to produce $100$ cups of coffee is $\$11.02$, while the cost to produce $400$ cups is $\$40.12$. Assume the cost is a linear function of $x$, the number of cups produced. $a.$ Find a formula for $C(x)$. $b.$ What is the fixed cost? $c.$ Find the total cost of producing $1000$ cups. $d.$ Find the total cost of producing $1001$ cups. $e.$ Find the marginal cost of the $1001$st cup. $f.$ What is the marginal cost of any cup and what does this mean to the manager?

Solution

VerifiedAssume that cost function has form: $C(x) = ax+b$

To produce 100 cups, cost is $\$11.02$. If we put it in the cost function we have:

$11.02=100a+b$

To produce 400 cups, cost is $\$40.12$. If we put it in the cost function we have:

$40.12=400a +b$

So we need to solve a linear system:

$\begin{gather*} 11.02 = 100a+b \\ 40.12 =400a+b \\ \end{gather*}$

Let's subtract second from the first equation:

$\begin{gather*} -29.1=-300a \\ \\ a = \dfrac{-29.1}{-300} = 0.097 \end{gather*}$

$\begin{gather*} b = 11.02 -100a = 11.02 -100 \cdot 0.097 = 1.32 \end{gather*}$

So the cost function is

$\boxed{C(x) = 0.097x + 1.32}$

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