Answer:
16 years
Step-by-step explanation:
Given
[tex]P(t) = t^3 - 24t^2 + 20t + 120[/tex]
Required
Year they made loss
[tex]P(t) = t^3 - 24t^2 + 20t + 120[/tex]
Start by differentiating P w.r.t t
Using first principle:
[tex]\frac{dP}{dt} = 3t^2 - 48t + 20[/tex]
Equate to 0, in order to solve for t
[tex]\frac{dP}{dt} = 0[/tex]
So, we have:
[tex]3t^2 - 48t + 20 = 0[/tex]
Solve using quadratic equation
[tex]t = \frac{-b \± \sqrt{b^2 - 4ac}}{2a}[/tex]
Where
a = 3, b = -48 and c = 20
So, we have:
[tex]t = \frac{-(-48) \± \sqrt{(-48)^2 - 4*3*20}}{2*3}[/tex]
[tex]t = \frac{-(-48) \± \sqrt{2064}}{2*3}[/tex]
[tex]t = \frac{-(-48) \± \45.4}{2*3}[/tex]
[tex]t = \frac{48 \± \45.4}{6}[/tex]
Split:
[tex]t = \frac{48 + \45.4}{6}[/tex] or [tex]t = \frac{48 - \45.4}{6}[/tex]
[tex]t = \frac{93.4}{6}[/tex] or [tex]t = \frac{2.6}{6}[/tex]
[tex]t = 15.56[/tex] or [tex]t = 0.43[/tex]
Approximate to whole numbers
[tex]t = 16[/tex] or [tex]t = 0[/tex]
Hence, they made loss 16 years after they took over
