Company QRS generated a net profit between 1992 and 1997 at a rate approximated by 20 t + 264.5 million dollars per year, where t is the time in years since 1992. Find the value, in 1992, of QRS's net profit over the 5-year period from 1992 to 1997. Assume that the interest rate is 2%, compounded continuously and round your answer to two decimal places.
Value: $ million
To find the value of QRS's net profit over the 5-year period from 1992 to 1997, we need to find the present value of each year's profit and then sum them up.
The formula for continuous compounding is:
PV = FV * e^(-rt)
where PV is the present value, FV is the future value, r is the interest rate, and t is the time in years.
First, we need to find the net profit for each year from 1992 to 1997:
For t=0 (1992): 20(0) + 264.5 = 264.5 million dollars
For t=1 (1993): 20(1) + 264.5 = 284.5 million dollars
For t=2 (1994): 20(2) + 264.5 = 304.5 million dollars
For t=3 (1995): 20(3) + 264.5 = 324.5 million dollars
For t=4 (1996): 20(4) + 264.5 = 344.5 million dollars
Next, we need to find the present value of each of these net profits:
PV(1992) = 264.5 * e^(-0.02*0) = 264.5 million dollars
PV(1993) = 284.5 * e^(-0.02*1) ≈ 278.71 million dollars
PV(1994) = 304.5 * e^(-0.02*2) ≈ 292.12 million dollars
PV(1995) = 324.5 * e^(-0.02*3) ≈ 303.97 million dollars
PV(1996) = 344.5 * e^(-0.02*4) ≈ 314.49 million dollars
Finally, we sum up the present values to find the total value:
Value = PV(1992) + PV(1993) + PV(1994) + PV(1995) + PV(1996)
Value ≈ 264.5 + 278.71 + 292.12 + 303.97 + 314.49
Value ≈ 1453.79 million dollars
The value in 1992 of QRS's net profit over the 5-year period from 1992 to 1997 is approximately 1453.79 million dollars.
To find the value of QRS's net profit over the 5-year period from 1992 to 1997, we need to find the definite integral of the rate of net profit function.
The rate of net profit function is given as 20t + 264.5 million dollars per year, where t is the time in years since 1992.
The integral of the rate of net profit function over the interval [0, 5] (since we want to find the net profit from 1992 to 1997) will give us the net profit over the 5-year period.
∫ (20t + 264.5) dt, from 0 to 5
= [10t^2 + 264.5t] from 0 to 5
= 10(5)^2 + 264.5(5) - 0
= 250 + 1322.5
= 1572.5 million dollars
Therefore, the value of QRS's net profit over the 5-year period from 1992 to 1997 is approximately $1572.5 million.
To find the value of QRS's net profit over the 5-year period from 1992 to 1997, we need to calculate the definite integral of the given function over the interval [0, 5].
The given function that represents the net profit is 20t + 264.5 million dollars per year, where t is the time in years since 1992.
To calculate the definite integral, we use the formula:
∫ [a, b] f(t) dt = F(b) - F(a)
where F(t) is the antiderivative of f(t).
First, let's find the antiderivative of the function 20t + 264.5.
The antiderivative of 20t is 10t^2, and the antiderivative of 264.5 is 264.5t.
Therefore, the antiderivative of the function 20t + 264.5 is:
F(t) = 10t^2 + 264.5t.
Now, we can calculate the definite integral over the interval [0, 5]:
∫ [0, 5] (20t + 264.5) dt = F(5) - F(0)
Substituting the values, we get:
F(5) = 10(5)^2 + 264.5(5) = 10(25) + 264.5(5) = 250 + 1322.5 = 1572.5
F(0) = 10(0)^2 + 264.5(0) = 0 + 0 = 0
Now, we can calculate the net profit over the 5-year period:
Net Profit = F(5) - F(0) = 1572.5 - 0 = 1572.5 million dollars.
So, QRS's net profit over the 5-year period from 1992 to 1997 is approximately $1572.5 million.