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Question 238667: Marginal Cost of Coffee. 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 C(x) 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 1001st cup.
f. What is the marginal cost of any cup and what does this mean to the manager?
I would appreciate any help at all if possible. This is really getting on my nerves. Thanks to who may help me.
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website!
Since it is given that ) is linear, and you are given two points on the Cups(x) vs. Cost(y) graph, use the two-point form of the equation of a line and then put it into slope-intercept form to derive the definition of
(x\ -\ x_1) )
Where ) and ) are the coordinates of the given points. For this problem ) and )
(x\ -\ 100) )
You can do your own arithmetic to verify, but the slope is 0.097 and the y-intercept, which is equal to the fixed cost in this example is 1.32.
So the slope-intercept, or function of  form is:
Substitute 1000 for and calculate, then substitute 1001 for and calculate, then calculate the difference. Given proper arithmetic since this is a linear function, the marginal value at 1001 will be the same as the marginal value at any other value of the independent variable, namely the value of the slope.
Marginal value is generally defined as the first derivative of the total cost function. But in the case of a linear function, the first derivative and the slope are the same thing.
It means that it costs him less than a penny for each cup of coffee he serves after he as recouped the fixed cost of $1.32.
John
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