Finally, the firm opened in 2000. It has determined that the overall firm\'s rev
ID: 1201582 • Letter: F
Question
Finally, the firm opened in 2000. It has determined that the overall firm's revenue over the past 16 years can be estimated by F(t) = -0.087t^3 + 1.271t^2 + 15.45t + 108.5 where t represents the year with t = 0 corresponding to 2000 and F(t) being revenue in 10,000 dollars. Using derivatives show that the firm's revenue was increasing from 2000 to 2013. Find when the revenue reached the maximum amount from 2000 to 2016. Find when the revenue was increasing at the fastest rate using the second derivative. In Excel, graph the revenue function F (t) and the first and second derivatives of the revenue functions using at least 50 data points over the domain of the 16 years. Label the graph accordingly. All key values you found through derivative work should be visable in your graph.Explanation / Answer
F = - 0.087t3 + 1.271t2 + 15.45t + 108.5
(1) Revenue function is increasing when dF / dT > 0
dF / dt = - 0.261t2 + 2.542t + 15.45
When t = 0, dF / dt = 15.45 > 0
When t = 13, dF / dt = (- 0.261 x 13 x 13) + (2.542 x 13) + 15.45 = - 44.109 + 33.046 + 15.45 = 4.387 > 0
So, total revenue is increasing between 2000 and 2013 (t = 0 and t = 13).
(2) Total revenue is maximum when dF / dt = 0
- 0.261t2 + 2.542t + 15.45 = 0
0.261t2 - 2.542t - 15.45 = 0
This is a quadratic equation, solving which (using online quadratic solver):
t = 13.9 or t = - 4
So, total revenue is highest in year 14.
(3)
dF2 / d2t = - 0.522t + 2.542
Rate of increase is highest when dF2 / d2t = 0
- 0.522t + 2.542 = 0
0.522t = 2.542
t = 4.87
So rate of increase was highest in year 5.
NOTE: First 3 questions are answered.
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