Christine's Creamery sells tubs of homemade vanilla ice cream. The total daily profit, in dollars, can be modeled by the function P(x)=–4x2+88x–200, where x is the price of one tub of ice cream, also in dollars.Complete the square to rewrite the function in the form P(x)=a(x–h)2+k. What are the values of a, h, and k?a=h=k=Now use your answers to complete the sentence.The maximum daily profit is $. The creamery can make this amount by charging $ per tub.Submit
Question
Christine's Creamery sells tubs of homemade vanilla ice cream. The total daily profit, in dollars, can be modeled by the function P(x)=–4x2+88x–200, where x is the price of one tub of ice cream, also in dollars.Complete the square to rewrite the function in the form P(x)=a(x–h)2+k. What are the values of a, h, and k?a=h=k=Now use your answers to complete the sentence.The maximum daily profit is per tub.Submit
Solution
To rewrite the function in the form P(x)=a(x–h)²+k, we first need to complete the square.
The given function is P(x) = -4x² + 88x - 200.
First, factor out a -4 from the x² and x terms:
P(x) = -4(x² - 22x) - 200.
To complete the square, we need to add and subtract (b/2)² inside the parentheses, where b is the coefficient of x, which is -22 in this case. So, (b/2)² = (-22/2)² = 121.
P(x) = -4[(x² - 22x + 121) - 121] - 200 = -4[(x - 11)² - 121] - 200 = -4(x - 11)² + 484 - 200 = -4(x - 11)² + 284.
So, the function in the form P(x)=a(x–h)²+k is P(x) = -4(x - 11)² + 284.
Therefore, a = -4, h = 11, and k = 284.
The maximum daily profit is 11 per tub.
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