If P dollars are invested at an annual rate of r%, then in t years this investment grows to F dollars, where $F = P \left( 1 + \frac { r } { 100 } \right) ^ { t }$ .If you solve this equation for P and hold F and r constant, what will the sign of $\frac { d P } { d t }$ be?
A)positive
B)negative

Question

Quizplus · Accepted Answer

To solve for $P$, we rearrange the given equation to $P = \frac{F}{\left(1 + \frac{r}{100}\right)^t}$. Differentiating both sides with respect to $t$, we get $\frac{dP}{dt} = -\frac{F \cdot \ln\left(1 + \frac{r}{100}\right)}{\left(1 + \frac{r}{100}\right)^t}$, which is negative because $F$, $r$, and the base of the natural logarithm are positive.

If P Dollars Are Invested at an Annual Rate of R