pimd-like math problem

Derivatives are just slopes. If a line is y=3x then it's derivative or slope is 3. If a line is y=x^2 then the slope can be modeled as 2x for any x value you plug in. Note that if you have a graph that has a slope that increases then decreased then you have a maximum. Notice the vertex of y=-x^2 5: it has a maximum at x=0. Note that it's slope changes from positive to negative at x=0, this tells you that you have a maximum. This means that when the derivative or slope is zero, you usually have a max or min (few exceptions) so for this optimization problem I have posted, you need come up with an equation to model the expressions and take the derivative of that equation and set it equal to zero to find either a max or minimum which will then make it very easy to find your answer.

There is unicorns math lesson 
 
A graphing calculator would spit out the answer without having to take a derivative. If you come up with the right equation and plug it in the graphing calculator then you can look at the graph to find a maximum. To find the maximum by hand without plugging in every single value that you can think of, you set the derivative equal to zero as previously stated. I would like to add that your approach of 4.96m and 21% and then looking other numbers and their effect is a good one. You have to be able to come up with equations to relate your data and by doing the approach that you did, should make coming up with the equation easier.
 
I'm bout to do this. I'll be right back. Need pencil. Need paper. Need an online graphing calculator.
 
Coming up with the equation is the hardest part. You want to figure out what you are solving for and then break it down piece by piece, you can write other equations off to the side to try to help you model what you need and figure out your one big equation.
 
I came up with Y=(5,000,000 - 40,000x)(.2 plus .01x)

If that's right I should be able to find out what I'd make per item and the maximum of it's graph after I distribute and get my quadratic should be the best item to profit combination.
 
Ha. Thanks, but don't ask about part 2. That's beyond the 11th grade. I wish I hadn't been conned out of taking calculus. 
 
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I've always found math easy. My guidance councilor wouldn't let me take calculus this year because "my schedule would be too loaded."

I'm taking college history and college English. Those are "gimme" classes as in "gimme my free college credits." now I'm only one year ahead. When I could have finished highschool this year.
 
Part 2's equation is very similar except it involves N in the equation. It still has X in the equation. You come up with the equation almost the same way too. You have to understand that if N = 1 then you should get the same answer that you got in part 1. Therefore when building your equation, you should be able to look at it and see that if you plug in N=1 then you have the exact same equation as part one. Also imagine what your profit would be if it was your 2nd hit, imagine what it would be for your 3rd hit and then try to come up with an equation to relate them 
 
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The first one was me messing up the second half.

The second is my correction.

The extra "Y=" is me being lazy and choosing not to distribute. 
 
I solved mine earlier by distributing but I used the product rule to avoid distributing and it saved a lot of work. The product rule is for taking the derivatives of two numbers that are multiplied together
 
I still don't understand this derivative business but I think I'll make it in my crappy math for college readiness class. 
 
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