caleb has $2,500 in his savings account after 18 months. the account earns an annual interest rate of 3.8%…

Explanation:

Step1: Identify the compound - interest formula

The compound - interest formula when compounded $n$ times a year is $A = P(1+\frac{r}{n})^{nt}$, where $A$ is the final amount, $P$ is the principal (initial amount), $r$ is the annual interest rate (in decimal form), $n$ is the number of times compounded per year, and $t$ is the number of years. Given $A = 2500$, $r=0.038$ (since $3.8%=0.038$), $n = 365$ (compounded daily), and $t=\frac{18}{12}=1.5$ years. We need to solve for $P$.

Step2: Rearrange the formula for $P$

From $A = P(1+\frac{r}{n})^{nt}$, we can get $P=\frac{A}{(1 +\frac{r}{n})^{nt}}$. Substitute the values: [ \begin{align*} P&=\frac{2500}{(1+\frac{0.038}{365})^{365\times1.5}}\ &=\frac{2500}{(1 + 0.00010411)^{547.5}}\ &=\frac{2500}{(1.00010411)^{547.5}} \end{align*} ] Calculate $(1.00010411)^{547.5}\approx1.0587$. Then $P=\frac{2500}{1.0587}\approx2361.48$.

Answer:

$$2361.48$