What Price Maximizes Profit for Snax-Treat Sales at the Corner Store?

[cscott]

The local corner store currently sells 240 bags weekly of Snax-Treat at a price of $3.29 each. Sales predictions indicate that each 25 cent decrease in price will increase sales by 60 bags weekly. If the store pays $2.00 for each bag, what prices will maximize profit?

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P(x) = (240 + 60x)[(3.29 - 2.00) - 0.25x] = -15x^2 + 17.4x + 309.6

A maximum at 0.58. If I round up to 1 beacuse x \epsilon N then the price must be 3.04 to maximize profit?

 

[Hurkyl]

I don't think you have to assume that x is an integer.

But if you do, rounding is the right thing to do -- but the reason is nontrivial. Can you explain why? (p.s. I'm assuming you found the maximum correctly)

 

[cscott]

Now that you mention it, and after I re-read the question, it isn't that great of an assumption.

 

[HallsofIvy]

The local corner store currently sells 240 bags weekly of Snax-Treat at a price of $3.29 each. Sales predictions indicate that each 25 cent decrease in price will increase sales by 60 bags weekly. If the store pays $2.00 for each bag, what prices will maximize profit?

--

P(x) = (240 + 60x)[(3.29 - 2.00) - 0.25x] = -15x^2 + 17.4x + 309.6

A maximum at 0.58. If I round up to 1 beacuse x \epsilon N then the price must be 3.04 to maximize profit?

One thing I strongly recommend you do is write down explicitely what your variable represents! It was not immediately obvious to me from your equation what x represents nor why x should be an integer (especially if you get x= 0.58 for a maximum!).

I think, if I interpret your equation correctly, that x is the number of "60 bag" increases in sale per week and so also the number of "25 cent" reductions in price. If you got x=0.58 for this, then that means a reduction in price of 0.58(25)= 14.5 with an increase in sales of 0.58(60)= 34.8. You might want to check whether a reduction in price of 14 or 15 cents doesn't give the maximum.