A store owner can sell 275 key chains per day at a price of $2 each. The cost per key chain to the owner is $1.50. For each 10 cent increase in price, there is a decrease in sales of 25 key chains. What is her maximum daily profit
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Let x be the amount of 10 cent price increases.
Then, profit(x) = p(x) = (275-25x)[2(1 + .1x) - 1.5] = -5x^2 + 42.5x +137.5.
We take the derivative of p(x) to find the maximum value of x:
p'(x) = -10x + 42.5.
Set p'(x) = 0 and solve for x:
x = 4.25.
So, the maximum daily profit is p(4.25) = $677.34.
Then, profit(x) = p(x) = (275-25x)[2(1 + .1x) - 1.5] = -5x^2 + 42.5x +137.5.
We take the derivative of p(x) to find the maximum value of x:
p'(x) = -10x + 42.5.
Set p'(x) = 0 and solve for x:
x = 4.25.
So, the maximum daily profit is p(4.25) = $677.34.