2500 hot dogs must be sold to make a profit
If they can sell 5000 hot dogs at $4 per dog, the selling price for the 5000dogs will be expressed as:
SP= 4 * 5000
SP = 4 + 0.25x
Total number hot dog sold = 5000 - 200x
Revenue = Sales price*Total number hot dog sold
When the price is increased by $0.25 then the revenue will be expressed as [tex]g(x) = (4+0.25x) (5000 - 200x)[/tex]
Taking the product, the resulting function will be;
[tex]g(x) = -50x^2 + 1250x + 20000[/tex]
To get the maximum profit, we need to maximize the revenue by differentiating the function and equating it to zero to have:
[tex]g'(x) = -100x + 450\\0=-100x + 1250\\100x = 1250\\x = 12.5[/tex]
Substittute x = 12.5 into the total number hot dog sold function
Recall that n = 5000 - 200x
n = 5000 - 200(12.5)
n = 5000 - 2500
n = 2500
Hence 2500 hot dogs must be sold to make a profit
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