If I can sell a chip for 4 dollars and know the marginal cost function for making x chips a day to be C'(x)=4/(√(1+0.04x)) with the limit of having to make fewer than 400 chips, what is the profit for making and selling 200 chips given that I can see each chip for 4 dollars? I also know C(200) is equal to 800 which means I can find any fixed costs and I believe them to be 200 but just am a bit confused on the switching to profit part.

Question

If I can sell a chip for 4 dollars and know the marginal cost function for making x chips a day to be C'(x)=4/(√(1+0.04x)) with the limit of having to make fewer than 400 chips, what is the profit for making and selling 200 chips given that I can see each chip for 4 dollars? I also know C(200) is equal to 800 which means I can find any fixed costs and I believe them to be 200 but just am a bit confused on the switching to profit part.

Answer 1

C'(x)=4/(√(1+0.04x))

so the total cost for making x chips is
C(x) = 40√(x+25) + k
since C(200) = 800,
800 = 40*15 + k
k = 200
revenue for selling x is 4x
the profit p(x) = revenue - cost, so
p(200) = 4*200 - 800 = 0

Answer 2

So 200 would be the fixed cost then since it is K?

Answer 3

nevermind I figured it out

Answer 4

To find the profit for making and selling 200 chips, we need to subtract the total cost from the total revenue.

The total cost (C) can be calculated by integrating the marginal cost function (C'(x)) with respect to x. In this case, the integral of C'(x) will give us the total cost function (C(x)).

Since we are given that C(200) is equal to 800, we can use this information to determine any fixed costs. Since C(200) represents the total cost of making and selling 200 chips, it includes both the variable costs (dependent on the number of chips produced) and the fixed costs (independent of the number of chips produced).

So, we can write the total cost function as C(x) = fixed costs + variable costs.

Since we are considering the fixed costs to be 200, we can rewrite the total cost function as C(x) = 200 + variable costs.

Since C(200) is equal to 800, we can substitute these values into the total cost function to solve for the variable costs:

800 = 200 + variable costs
variable costs = 800 - 200
variable costs = 600

Now that we have determined the variable costs, we can calculate the total cost for producing 200 chips:

C(200) = 200 + variable costs
C(200) = 200 + 600
C(200) = 800

The total cost for making and selling 200 chips is 800.

To find the total revenue, we multiply the selling price per chip (4 dollars) by the number of chips sold (200):

Total revenue = 4 * 200
Total revenue = 800

Now, we can calculate the profit by subtracting the total cost from the total revenue:

Profit = Total revenue - Total cost
Profit = 800 - 800
Profit = 0

Therefore, the profit for making and selling 200 chips is 0 dollars.