Solution
Step 1
Write the compound interest formula
[tex]\text{A = P\lparen1 + }\frac{r}{n})^{nt}[/tex]
Step 2
n = 4 (quarterly)
[tex]\begin{gathered} \text{P = x} \\ \text{A = 2x} \\ r\text{ = 7}\frac{3}{4}\text{ = 7.75\% = 0.0775} \end{gathered}[/tex]
Step 3:
Substitute in the formula to find t.
[tex]\begin{gathered} 2x\text{ = x\lparen 1 + }\frac{0.0775}{4})^{4t} \\ \text{2 = \lparen1 + 0.019375\rparen}^4t \\ \text{2 = 1.019375}^{4t} \\ Take\text{ natural logarithm of both sides} \\ In(2)\text{ = 4t In\lparen1.019375\rparen} \\ 4t\text{ = }\frac{ln(2)}{ln(1.019375)} \\ 4t\text{ = 36.12080351} \\ t\text{ = }\frac{36.12080351}{4} \\ t\text{ = 9.03 years} \end{gathered}[/tex]
Final answer
t = 9.03
