The total expenses (or costs) for the concert are given by the model:
C(x)=5x+6
Where x is the ticket price
The income (or revenue) is modeled by the function:
[tex]I\mleft(x\mright)=-x^2+20x-30[/tex]a) The profit is calculated as the incomes minus the costs:
P(x) = I(x) - C(x)
Substituting the above models:
[tex]\begin{gathered} P(x)=-x^2+20x-30-(5x+6) \\ \text{Operating:} \\ P(x)=-x^2+20x-30-5x-6 \\ P(x)=-x^2+15x-36 \end{gathered}[/tex]b) To calculate the break-even point, we equate the profit to zero:
[tex]\begin{gathered} -x^2+15x-36=0 \\ \text{Multiplying by -1} \\ x^2-15x+36=0 \end{gathered}[/tex]The polynomial can be factored:
( x - 12 ) ( x - 3 ) = 0
We have two solutions:
x=12, x=3
There are two break-even points, when the price is $3 or when the price is $12
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