the value of an investment is expected to grow by 5% per year. how long will it take for the value to…

Explanation:

Step1: Set up compound - growth formula

Let the initial investment be $P$, the final value be $A$, the annual interest rate $r = 0.05$, and the number of years be $t$. The compound - growth formula is $A=P(1 + r)^t$. We want $A = 2P$, so $2P=P(1 + 0.05)^t$.

Step2: Simplify the equation

Divide both sides of the equation $2P=P(1 + 0.05)^t$ by $P$ (since $P\neq0$), getting $2=(1.05)^t$.

Step3: Take the natural logarithm of both sides

$\ln(2)=\ln(1.05^t)$. Using the property of logarithms $\ln(a^b)=b\ln(a)$, we have $\ln(2)=t\ln(1.05)$.

Step4: Solve for $t$

$t=\frac{\ln(2)}{\ln(1.05)}$. Since $\ln(2)\approx0.693$ and $\ln(1.05)\approx0.0488$, then $t=\frac{0.693}{0.0488}\approx14.2$.

Answer:

about 14 years