A chemical manufacturer sells sulfuric acid in bulk at a price of...

Question

A chemical manufacturer sells sulfuric acid in bulk at a price of $400 per unit. If the daily total production cost in dollars for x units is:

C(x)=100,000+200x+0.01x²

and if the daily production capacity is at most 7000 units, how many units of sulfuric acid must be manufactured and sold daily to maximize the profit?

Answer 1

C(x) is cost

R(x) is revenue

P(x) is profit = R(x)-C(x)

C(x) is given

R(x)=400x

Therefore P(x)=-100,000+200x-0.01x^2

Take the derivative of P(x) with respect to x. dP/dx=200-0.02x

When dP/dx=0 there is a maximum or a minimum.  X is equal to 10000 at this point.

Taking the derivative again d²P/dx²=-0.02 which indicates that it is a maximum. However, this point is above the daily limit for production.

Using the first derivative once more

 dP/dx=200-.02x

for all numbers 0<=x<=7000 the slope is positive. This indicates that the profit is always increasing in this range.

Therefore the highest profit will be at 7000 units.