The future value that accrues when $900 is invested at 7%, compounded continuously, is , where t is the number of years. At what rate is the money in this account growing when
A) $18.12 per year
B) $69.63 per year
C) $1812.38 per year
D) $965.26 per year
E) $126.87 per year

Question

The future value that accrues when $900 is invested at 7%, compounded continuously, is  , where t is the number of years. At what rate is the money in this account growing when 
A) $18.12 per year 
B) $69.63 per year 
C) $1812.38 per year 
D) $965.26 per year 
E) $126.87 per year

Quizplus · Accepted Answer

To find the rate at which the money is growing, we need to take the derivative of the future value formula with respect to time:

$V(t) = P e^{rt}$ where $P = 900$ and $r = 0.07$ (continuously compounded)

$\frac{dV}{dt} = Pr e^{rt}$

We want to find the rate at which the money is growing when $V = 11eaa894_64ad_0c57_a158_e9fba5516bfa_TB6775_11$, which means we need to solve for $t$:

$11eaa894_64ad_0c57_a158_e9fba5516bfa_TB6775_11 = P e^{rt}$

$\ln (11eaa894_64ad_0c57_a158_e9fba5516bfa_TB6775_11 / P ) = rt$

$t = \frac{1}{r} \ln (11eaa894_64ad_0c57_a158_e9fba5516bfa_TB6775_11 / P )$

Now we can substitute this value of $t$ into our expression for $\frac{dV}{dt}$:

$\frac{dV}{dt} = Pr e^{rt} = P r e^{r (\frac{1}{r} \ln (11eaa894_64ad_0c57_a158_e9fba5516bfa_TB6775_11 / P ))}$

Simplifying:

$\frac{dV}{dt} = r \cdot 11eaa894_64ad_0c57_a158_e9fba5516bfa_TB6775_11 \approx \boxed{	extbf{(E)}\ 126.87\ 	ext{per year}}$

The Future Value That Accrues When $900 Is Invested at 7