Answer :
31 years
Explanation:We would apply the compound interest forula:
[tex]A=P\mleft(1+\frac{r}{n}\mright)^{nt}[/tex]A = future amount = $1000
P = principal = $250
r = rate = 4.5% = 0.045
n = compounded quarterly = 4 times
n = 4
t = time = ?
Inserting the values into the formula:
[tex]\begin{gathered} 1000\text{ = 250(1 + }\frac{0.045}{4})^{4\times t} \\ 1000=250(1+0.01125)^{4t} \\ \text{divide through by 250} \\ \frac{1000}{250}=\text{ }(1+0.01125)^{4t} \\ 4\text{ = (1}.01125)^{4t} \end{gathered}[/tex][tex]\begin{gathered} \text{Taking log of both sides:} \\ \log 4=log(1.01125)^{4t} \\ \log 4=4t\lbrack log(1.01125)\rbrack \\ 0.6021\text{ = 4t(}0.0049) \end{gathered}[/tex][tex]\begin{gathered} 0.6021\text{ = }0.0196t \\ \text{divide both sides by 0.0196} \\ \frac{0.6021}{0.0196}=\frac{0.0196t}{0.0196} \\ 30.72\text{ = t} \\ To\text{ the nearest whole number, t = 31 years} \end{gathered}[/tex]It takes 31 years to reach $1000.
