is this correct?

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 is a linear function of x, the number
of cups produced.

a. Find a formula for C(x).
(100,11.02) and (400,40.12)
slope is: 0.291
therefore using point (100,11.02)
I end up: y = 0.291x -18.08

b. What is the fixed cost?
the cost is -18.08

c. Find the total cost of producing 1000 cups.

c(1000)= 0.291(1000)-18.08 = 272.92

d. Find the total cost of producing 1001 cups.

c(1001)=0.291(1001)-18.08= 273.211

e. Find the marginal cost of the 1001st cup.
I do not know this part

f. What is the marginal cost of any cup and what does this
mean to the manager?

I don't know what to do here

On a. How did you get that slope? Isn't slope=(40.12-11.02) /(400-100)

but if i do slope with your points i get a -0.247 etc.....

Hmmmm. Check your math, in round terms, that is 30 divided by 300. That is not one fourth, not even close.

y=.097x+1.32

is the correct formula,
the rest should fall into place!

You are correct, my apologies for the incorrect calculation.

To find the slope, you should use the formula:

slope = (y2 - y1) / (x2 - x1)

Using the given points (100, 11.02) and (400, 40.12), the calculation would be:

slope = (40.12 - 11.02) / (400 - 100)
slope = 29.1 / 300
slope ≈ 0.097

Therefore, the correct formula for C(x) is:

C(x) = 0.097x + b

To find the value of the constant term (b), you can substitute the coordinates of one of the points into the equation.

Using (100, 11.02):

11.02 = 0.097(100) + b
11.02 = 9.7 + b
b ≈ 1.32

So, the correct formula for C(x) is:

C(x) = 0.097x + 1.32

To find the slope in order to determine the formula for C(x), you need to use the formula for calculating slope, which is:

slope = (change in y) / (change in x)

In this case, the change in y is the difference in cost, and the change in x is the difference in the number of cups produced. So, you are correct that the formula for slope is:

slope = (40.12 - 11.02) / (400 - 100)

Calculating this, you get:

slope = 29.10 / 300
≈ 0.097

So, it seems there was a calculation error in the original question when it stated the slope as 0.291. The correct slope is approximately 0.097.

With the correct slope, let’s recalculate the formula for C(x) using the point (100, 11.02):

C(x) = slope * x + b

11.02 = 0.097 * 100 + b

11.02 = 9.7 + b

b ≈ 1.32

Therefore, the correct formula for C(x) is:

C(x) = 0.097x + 1.32

Now, let's move on to the remaining questions:

b. The fixed cost is the y-intercept of the cost function, which is the value of C when x = 0. From the formula C(x) = 0.097x + 1.32, we can see that the fixed cost is approximately $1.32.

c. To find the total cost of producing 1000 cups, we substitute x = 1000 into the formula C(x):

C(1000) = 0.097 * 1000 + 1.32
= 97 + 1.32
= 98.32

Therefore, the total cost of producing 1000 cups is approximately $98.32.

d. To find the total cost of producing 1001 cups, we substitute x = 1001 into the formula C(x):

C(1001) = 0.097 * 1001 + 1.32
= 97.097 + 1.32
= 98.417

Therefore, the total cost of producing 1001 cups is approximately $98.417.

e. The marginal cost of the 1001st cup refers to the additional cost incurred by producing that specific cup. In this case, since the cost function is linear, the marginal cost is equal to the slope of the function. Therefore, the marginal cost of the 1001st cup is approximately $0.097.

f. The marginal cost of any cup is the increase in cost that results from producing one extra cup. In this case, the marginal cost is $0.097 for each cup produced. This is important for the manager as it helps them understand how costs change as they increase production. By knowing the marginal cost, the manager can make informed decisions about pricing, production levels, and maximizing profitability.