Evelyn is a graphic designer who runs an online T-shirt store. She figures out that her daily profit, in dollars, can be modeled by the function P(x)=–2x2+54x–280, where x is the price of each T-shirt, also in dollars. Evelyn will break even when her profit is $0.Rewrite the function in the form P(x)=a(x–r1)(x–r2) by factoring. What are the values of a, r1, and r2?a=r1=r2=Now use your answers to complete the sentence.Evelyn will break even by charging either $ or $. She will make a profit by charging a price between those two amounts.Submit
Question
Evelyn is a graphic designer who runs an online T-shirt store. She figures out that her daily profit, in dollars, can be modeled by the function P(x)=–2x2+54x–280, where x is the price of each T-shirt, also in dollars. Evelyn will break even when her profit is or $. She will make a profit by charging a price between those two amounts.Submit
Solution
To rewrite the function P(x) = -2x^2 + 54x - 280 in the form P(x) = a(x - r1)(x - r2), we first need to factor the equation.
The equation can be factored as follows:
P(x) = -2(x^2 - 27x + 140)
Now, we need to factor the quadratic equation inside the parentheses. The factors of 140 that add up to 27 are 20 and 7. So, we can write the equation as:
P(x) = -2(x - 20)(x - 7)
So, a = -2, r1 = 20, and r2 = 7.
Evelyn will break even by charging either 7. She will make a profit by charging a price between those two amounts.
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