Question 1153056


{{{p(x)=-3x^2+18x-4}}} where {{{p(x)}}} is the amount of profit in hundreds of thousands of dollars and {{{x }}}is the number of years of operation. 
He realizes his company is on the downturn and wishes to sell before he ends up in debt.

a. When will Mr. Unlucky’s business show the maximum profit? ___ years

{{{p(x)=-3x^2+18x-4}}} ...since you have an upside down parabola, max is at vertex; so, write your equation in vertex form

{{{p(x)=-3x^2+18x-4}}}....group {{{-3x^2+18x}}}, factor out {{{-3}}}

{{{p(x)=-3(x^2-6x)-4}}}.......complete square

{{{p(x)=-3(x^2-6x+b^2)-(-3)b^2-4}}}

{{{p(x)=-3(x^2-6x+3^2)+3*3^2-4}}}

{{{p(x)=-3(x-3)^2+27-4}}}

{{{p(x)=-3(x-3)^2+23}}}=>{{{h=3}}} and {{{k=23}}}

 the maximum profit is at {{{3}}} years

b. What is the maximum profit? ____ hundreds of thousands

the maximum profit is {{{23}}} hundreds of thousands

c. During what time period will Mr. Unlucky expect his business’ profits to be on the downturn? After {{{3}}} years