I have a marginal cost function decreasing as hats are made. The marginal cost for making x thousand hats in a day is C'(x)=40/√x in hundreds of dollars. I know 40,000 dollars are necessary to make 1,000 hats which essentially gives me a point at (1,40). If I'm trying to calculate the cost to make 16 thousand hats in a day, wouldn't my first step be to find the antiderivative and then plug in my given point to get the constant which in this case is K (since C is already used for the function). Then I can plug in 16 to get the value in hundreds?
When I try to get the antiderivative, I get 80x^(1/2)+C and then I go to try and plug in the point (1,40) to get C and I get -40 so then I assume I plug that and 16 in to get my answer in hundreds
Update, that gets 280 which is apparently wrong and I'm confused on if I messed up finding the antiderivative or if I messed up after?
To calculate the cost of making 16 thousand hats in a day, you're correct that you need to find the antiderivative of the marginal cost function first. The antiderivative of C'(x) = 40/√x is given by:
C(x) = ∫[40/√x] dx
To evaluate this integral, recall that the antiderivative of x^k is (1/(k+1))x^(k+1). Applying this rule to our function, we have:
C(x) = 40∫(1/√x) dx
Simplifying, we get:
C(x) = 40∫x^(-1/2) dx
Now, integrating x^(-1/2), we get:
C(x) = 40 * (2√x)
However, we are given the point (1, 40), which means when x = 1, C(x) = 40. Substituting these values into our equation, we can solve for the constant K:
40 = 40 * (2√1)
1 = 2
So, the value of the constant K is 2. Therefore, the cost function is:
C(x) = 40 * (2√x)
Now, to determine the cost of making 16 thousand hats in a day, you can simply plug in x = 16 into the cost function:
C(16) = 40 * (2√16)
Simplifying, we have:
C(16) = 40 * (2 * 4)
C(16) = 3200
Therefore, the cost to make 16 thousand hats in a day is $3200 in hundreds of dollars.