SOLUTION: How would I go about solving this? My first attempt did not seem right! Any help with this would be amazing, and if anyone knew the final solution for me to work towards would be s

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Question 1119978: How would I go about solving this? My first attempt did not seem right! Any help with this would be amazing, and if anyone knew the final solution for me to work towards would be so amazing.

This is an algebraic question.
The profit of a company can be modelled by the polynomial function P(t)=-4t^3+10t^2+8t-6, where P is the profit, in thousands of dollars, and t is the time, in years. When will the company make their maximum profit of $18 000?
Any help is so appreciated! Thank you.
Answer by ikleyn(52776) About Me  (Show Source):
You can put this solution on YOUR website!
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  Plot y = -4t%5E3%2B10t%5E2%2B8t-6 (red)  and y = 18 (green)



1.  Take the derivative  y'(t) = -12t^2 + 20t + 8.


2.  Equate it to zero:  -12t^2 + 20t + 8 = 0,  or, equivalently,

    3t^2 - 5t - 2 = 0


3.  Solve the quadratic equation  t%5B1%2C2%5D = %285+%2B-+sqrt%285%5E2+%2B+4%2A3%2A2%29%29%2F%282%2A3%29 = %285+%2B-+sqrt%2849%29%29%2F6.


    Take its positive root  t%5B1%5D = %285+%2B+7%29%2F6 = 2.


4.  Substitute t = 2 into your polynomial P(t) to check the value of the profit

    P(2) = -4%2A2%5E3%2B10%2A2%5E2%2B8%2A2-6 = 18.


Answer.   In two years.