The revenue, in millions of dollars, for a company in year t is given by the function:
R(t)=15*e^(0.08*t), 0≤t≤15
and the cost, in millions of dollars, to run the company in year t is approximated by:
C(t)=12*e^(−0.04*t), 0≤t≤15
where t is the number of years after January 1st of the year 2000. What was the net profit (in millions of dollars) for the company from January 1st in the year 2000 until January 1st in the year 2007? Round your answer to the nearest million dollars.
assuming profit = revenue - cost
P(x) = 15*e^(0.08*t) - 12*e^(−0.04*t)
P'(x) = 1.2e^(.08t) + .48e^(-.04t)
= 0 for a max of P(x)
1.2 e^ .08t + .48e^-.04t = 0
divide by .48
2.5e^.08t + e^-.04t = 0
e^-.48t( e^.56t + 1) = 0
I get no real solution for this, neither does Wolfram
http://www.wolframalpha.com/input/?i=e%5E(-.48t)(+e%5E(.56t)+%2B+1)+%3D+0
To find the net profit for the company from January 1st in the year 2000 until January 1st in the year 2007, we need to calculate the difference between the revenue and the cost over that time period.
First, let's find the revenue for January 1st, 2000, and January 1st, 2007.
For January 1st, 2000 (t=0), we can substitute t=0 into the revenue function:
R(0) = 15 * e^(0.08 * 0)
R(0) = 15 * e^0
R(0) = 15 * 1
R(0) = 15 million dollars
Similarly, for January 1st, 2007 (t=7), we can substitute t=7 into the revenue function:
R(7) = 15 * e^(0.08 * 7)
R(7) = 15 * e^(0.56)
R(7) ≈ 15 * 1.751
R(7) ≈ 26.27 million dollars
Next, let's find the cost for January 1st, 2000, and January 1st, 2007.
For January 1st, 2000 (t=0), we can substitute t=0 into the cost function:
C(0) = 12 * e^(-0.04 * 0)
C(0) = 12 * e^0
C(0) = 12 * 1
C(0) = 12 million dollars
Similarly, for January 1st, 2007 (t=7), we can substitute t=7 into the cost function:
C(7) = 12 * e^(-0.04 * 7)
C(7) = 12 * e^(-0.28)
C(7) ≈ 12 * 0.755
C(7) ≈ 9.06 million dollars
Now, we can calculate the net profit by subtracting the cost from the revenue:
Net Profit = R(7) - C(7)
Net Profit = 26.27 - 9.06
Net Profit ≈ 17.21 million dollars
Therefore, the net profit for the company from January 1st in the year 2000 until January 1st in the year 2007 is approximately 17.21 million dollars.