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Tanya runs a catering business. Based on her records, her weekly profit can be approximated by the function LaTeX: f\left(x\right)=2x^2-44x-150 f ( x ) = 2 x 2 − 44 x − 150 , where LaTeX: x x is the number of meals she caters. If LaTeX: f(x) f ( x ) is negative, it means that the business has lost money. What is the number of meals that Tanya needs to cater in order to break-even?

Respuesta :

The function

[tex] 2x^2-44x-150 [/tex]

is a parabola concave up, whose solutions are

[tex] 2x^2-44x-150=0 \iff x^2-22x-75=0 [/tex]

from here, you can use the quadratic formula

[tex] x_{1,2} = \dfrac{-b\pm\sqrt{b^2-4ac}}{2a} [/tex]

to find that the solutions of the parabola are [tex] -3,\ 25 [/tex]

So, the parabola is positive if [tex] x<-3 [/tex] (which wouldn't make sense in our case) or [tex] x<25 [/tex]

So, if Tanya caters 25 meals she breaks even, and starting with the 26th meal she will begin to profit.

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