Answer :
Final answer:
To find the present value of the income from the investment, calculate the discounted present value of the income stream using the formula provided. In this case, evaluate the integral and substitute the given values to find the present value of the income over the next 2 years. The present value is approximately $55,078.56.
Explanation:
To find the present value of the income from the investment, we need to calculate the discounted present value of the income stream. The formula for the present value of a continuous income stream is given by:
V = ∫ P(t) · e-rt dt
where V is the present value, P(t) is the income at time t, r is the interest rate, and e is the base of natural logarithm (approximately equal to 2.71828).
In this case, P(t) = 60000 - 300t and the interest rate r = 2%. Plugging in these values into the formula and integrating, we get:
V = ∫ (60000 - 300t) · e-0.02t dt = 30000(e-0.02t - 1)
To find the present value of the income over the next 2 years, we can evaluate the integral from t=0 to t=2:
V = ∫02 (60000 - 300t) · e-0.02t dt
Substituting the limits of integration and simplifying, we get:
V = 30000(e-0.04 - 1) + 30000(e-0.04 · 2 - e-0.04)
Simplifying further, we find that the present value of the income is approximately $55,078.56.
