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?

Question

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?

Alex · Answer

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

Alex · Answer

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?