$\text { Suppose the capitalized cost } C \text { is given by } C = C _ { 0 } + \int _ { 0 } ^ { n } c ( t ) e ^ { - n } d t \text {, where } C _ { 0 } \text { is the }$ original investment, $t$ is the time in years, $r$ is the annual interest rate compounded continuously, and $c ( t )$ is the annual cost of maintenance. Find the capitalized cost $C$ of an asset forever if $C _ { 0 } = 670,000 , c ( t ) = 22,000$, and $r = 0.09$. Round your answer to the nearest dollar.
A) $\$ 914,433$
B) $\$ 671,980$
C) $\$ 914,456$
D) $\$ 914,444$
E) $\$ 693,980$

Question

$\text { Suppose the capitalized cost } C \text { is given by } C = C _ { 0 } + \int _ { 0 } ^ { n } c ( t ) e ^ { - n  } d t \text {, where } C _ { 0 } \text { is the }$ original investment, $t$ is the time in years, $r$ is the annual interest rate compounded continuously, and $c ( t )$ is the annual cost of maintenance. Find the capitalized cost $C$ of an asset forever if $C _ { 0 } = 670,000 , c ( t ) = 22,000$, and $r = 0.09$. Round your answer to the nearest dollar.
A) $\$ 914,433$
B) $\$ 671,980$
C) $\$ 914,456$
D) $\$ 914,444$
E) $\$ 693,980$

Quizplus · Accepted Answer

Plugging in the given values for $C_0,c(t),$ and $r,$ we have
$$C=670000+\int_0^\infty22000e^{-0.09t}dt=670000+\frac{22000}{0.09}=914444.44$$
Rounding this to the nearest dollar gives us $\$914,444$, which is closest to option D.

Original Investment ttt Is the Time in Years rrr Is the Annual Interest Rate Compounded Continuously, And

Original Investment $t$ Is the Time in Years $r$ Is the Annual Interest Rate Compounded Continuously, And