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Answer :
Final answer:
To find the present value of the income from an investment over 2 years at a continual compounded interest rate, integrate the income function discounted by the exponential of the negative product of the interest rate and time. Calculate this definite integral from 0 to 2, and you will get the present value, which can be rounded to the nearest dollar.
Explanation:
To find the present value of the income from an investment with a revenue stream described by P(t) = 60000 + 300t dollars/year over the next 2 years at a continuous compounded interest rate of 2% per year, we can use the formula:
V = ∫ P(t)e^{-rt} dt from t = 0 to t = T
Where:
- V is the present value of the investment
- P(t) is the income rate as a function of time
- e is the base of the natural logarithm
- r is the annual interest rate (0.02 for 2%)
- t is time in years, and T is the total time period for the investment
Let's do the integration to find the present value:
V = ∫ (60000 + 300t)e^{-0.02t} dt from 0 to 2
Integrating, we get:
V = [-1,500,000 e^{-0.02t} - 15,000t e^{-0.02t}] from 0 to 2
Calculate the expression for t = 2 and t = 0, and subtract the latter from the former to find the value of the integral:
V = (-1,500,000 e^{-0.04} - 15,000*2 e^{-0.04}) - (-1,500,000 - 0)
V = -1,500,000 e^{-0.04} - 30,000 e^{-0.04} + 1,500,000
Now we can calculate the numerical value and round it to the nearest dollar.
After computing the numerical values and rounding,
The present value of the income from the investment is approximately $_______ . (Input the calculated value)
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