Suppose the annual rate of income from an investment at any time t is $K ( t ) = - 100 + 50 t$ . What is the formula for the present value of the income over the next 5 years at a 6% interest rate compounded continuously?
A) $\int _ { 0 } ^ { 5 } ( 50 t - 100 ) e ^ { 0.06 t } d t$
B) $\int _ { 0 } ^ { 5 } ( 50 t - 100 ) e ^ { - 0.06 t } d t$
C) $\int _ { 1 } ^ { 5 } ( 50 t - 100 ) e ^ { - 0.06 t } d t$
D) $\int _ { 1 } ^ { 5 } 50 t ^ { - 0.06 t } d t$
E) none of these

Question

Quizplus · Accepted Answer

The present value of income over time, considering continuous compounding, is calculated using the formula $PV = \int (Income) \cdot e^{-rt} dt$, where $r$ is the interest rate and $t$ is time. Here, the income $K(t) = -100 + 50t$, the interest rate $r = 0.06$, and the time period is from 0 to 5 years, making the correct formula $\int _ { 0 } ^ { 5 } ( 50 t - 100 ) e ^ { - 0.06 t } d t$.

Suppose the Annual Rate of Income from an Investment at Any