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Maximizing Profits

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I have a website where I offer cash rebates for buying stuff. For instance, if someone purchases a pair of shoes, I give them a $10 cash rebate. For that particular sale I will receive a commission of say, $25, so my net revenue from that offer is $15. The goal is to offer a rebate that is large enough that lots of people are incentivized to purchase the pair of shoes, but not so large that I will be giving most of my commissions away in the form of cash rebates. I have tested the number of shoes sold at 2 different cash rebates so I have two sets of coordinates for the demand function where x is the payout and y is the number of shoes sold.
They are (5,15) and (10,40). We can assume that the slope is linear.
I want to find the size of the cash rebate that maximizes total profits.
How would I do this?

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I have a website where I offer cash rebates for buying stuff. For instance, if someone purchases a pair of shoes, I give them a $10 cash rebate. For that particular sale I will receive a commission of say, $25, so my net revenue from that offer is $15. The goal is to offer a rebate that is large enough that lots of people are incentivized to purchase the pair of shoes, but not so large that I will be giving most of my commissions away in the form of cash rebates. I have tested the number of shoes sold at 2 different cash rebates so I have ...

Solution Summary

The solution calculates size of rebate that maximizes total profits.

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