investment c: a bank account earns 5% per year, on average. how long will it take for an investment at this…

Explanation:

Step1: Recall compound - interest formula for doubling

The compound - interest formula for the future value of an investment is $A = P(1 + r)^t$, where $A$ is the future value, $P$ is the principal amount, $r$ is the annual interest rate, and $t$ is the number of years. When the investment doubles, $A = 2P$. So, $2P=P(1 + r)^t$. Canceling out $P$ (since $P\neq0$), we get $2=(1 + r)^t$.

Step2: Identify the interest rate

The annual interest rate $r = 0.05$ (since $5%=0.05$).

Step3: Solve for $t$

Take the natural logarithm of both sides of the equation $2=(1 + 0.05)^t$. We have $\ln(2)=\ln((1.05)^t)$. Using the property of logarithms $\ln(a^b)=b\ln(a)$, we get $\ln(2)=t\ln(1.05)$. Then $t=\frac{\ln(2)}{\ln(1.05)}\approx\frac{0.6931}{0.0488}\approx14.2$.

Answer:

It will take approximately 14.2 years for the investment to double.