Maximum Profit Problem

[kss]

In marketing a certain item, a business has discovered that the demand for the item is represented by


x = 1225/p^2


The cost of producing x items is given by C = 0.3x + 1200. Find the price per unit that yields a
maximum profit.

So far what I have is:

I rearranged the function to: p=35/(sqrt(x))

P= R - C
P= xp - (0.3x+1200)
P= x*(35/(sqrt(x))) - (0.3x + 1200)
P= 35*(sqrt(x)) - 0.3x + 1200

P'=(35)/(2*(sqrt(x))) - 0.3

Now I set P' = 0, but I cannot not figure out how to solve for x. Its been over an hour..

(35)/(2*(sqrt(x))) - 0.3 = 0

If someone can help me solve for x, I know that would give me the number to figure out the price per unit for maximum profit.

 

[HallsofIvy]

In marketing a certain item, a business has discovered that the demand for the item is represented by


x = 1225/p^2


The cost of producing x items is given by C = 0.3x + 1200. Find the price per unit that yields a
maximum profit.

So far what I have is:

I rearranged the function to: p=35/(sqrt(x))

P= R - C
P= xp - (0.3x+1200)
P= x*(35/(sqrt(x))) - (0.3x + 1200)
P= 35*(sqrt(x)) - 0.3x + 1200

P'=(35)/(2*(sqrt(x))) - 0.3

Now I set P' = 0, but I cannot not figure out how to solve for x. Its been over an hour..

(35)/(2*(sqrt(x))) - 0.3 = 0

If someone can help me solve for x, I know that would give me the number to figure out the price per unit for maximum profit.

It's strange that you can do all that Calculus and not solve a simple equation.

\(\displaystyle \frac{35}{2\sqrt{x}}- 0.3= 0\)

Add 0.3 to both sides
\(\displaystyle \frac{35}{2\sqrt{x}}= 0.3\)

Multiply both sides by \(\displaystyle 2\sqrt{x}\)
\(\displaystyle 35= 0.6\sqrt{x}\)

Divide both sides by 0.6
\(\displaystyle \frac{350}{6}= \sqrt{x}\)

Square both sides
\(\displaystyle \left(\frac{350}{6}\right)^2= x\)