To break even for her initial expenses, Cara must give at least 50 massages as per the quadratic equation of profit and losses modeled by her.
It is given to us that Cara has modeled her initial profit and losses by the formula
f(x) = -0.1944x² + 29.16x - 972
where,
x = number of 30-minute massages
We have to find out the number of massages she must give to break even for her initial expenses.
The given equation f(x) = -0.1944x² + 29.16x - 972 is in the form of a quadratic equation ax² + bx + c.
The roots of this quadratic equation is given by
[tex]x_{1} =\frac{-b+\sqrt{b^{2} -4ac} }{2a}[/tex] ----- (1)
and, [tex]x_{2} =\frac{-b-\sqrt{b^{2} -4ac} }{2a}[/tex] ----- (2)
From the given quadratic equation f(x) = -0.1944x² + 29.16x - 972, we can find out the values of a, b, and c such that -
a = -0.1944
b = 29.16
c = - 972
Substituting these values of a, b, and c in equations (1) and (2), we have
[tex]x_{1} =\frac{-b+\sqrt{b^{2} -4ac} }{2a}\\= > x_{1} =\frac{-29.16+\sqrt{29.16^{2} -4*(-0.1944)*(- 972)} }{2(-0.1944)}\\= > x_{1} =\frac{-29.16+\sqrt{850.30-755.82} }{-0.3888}\\ = > x_{1} =\frac{-29.16+9.72}{-0.3888} \\= > x_{1} =\frac{-19.44}{-0.3888} \\= > x_{1} =50[/tex]
and,
[tex]x_{2} =\frac{-b-\sqrt{b^{2} -4ac} }{2a}\\= > x_{2} =\frac{-29.16-\sqrt{29.16^{2} -4*(-0.1944)*(- 972)} }{2(-0.1944)}\\= > x_{2} =\frac{-29.16-\sqrt{850.30-755.82} }{-0.3888}\\ = > x_{2} =\frac{-29.16-9.72}{-0.3888} \\= > x_{2} =\frac{-38.88}{-0.3888} \\= > x_{2} =100[/tex]
Thus, Cara may give 50 or 100 massages. To break even for her initial expenses, Cara must give at least 50 massages as per the quadratic equation of profit and losses modeled by her.
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