Answer:
Explanation:
A) We'll use the below compound interest formula to solve the given problem;
[tex]A=P(1+r)^t[/tex]where P = principal (starting) amount
A = future amount = 2P
t = number of years
r = interest rate in decimal = 7.8% = 7.8/100 = 0.078
Since the interest is compounded weekly, then r = 0.078/52 = 0.0015
Let's go ahead and substitute the above values into the formula and solve for t;
[tex]\begin{gathered} 2P=P(1+0.0015)^t \\ \frac{2P}{P}=(1.0015)^t \\ 2=(1.0015)^t \end{gathered}[/tex]Let's now take the natural log of both sides;
[tex]\begin{gathered} \ln 2=\ln (1.0015)^t \\ \ln 2=t\cdot\ln (1.0015) \\ t=\frac{\ln 2}{\ln (1.0015)} \\ t=462.44\text{ w}eeks \\ t\approx\frac{462.55}{52}=8.89\text{ years} \end{gathered}[/tex]We can see that it will take 8.89 years for
B) when r = 13% = 13/100 = 0.13
Since the interest is compounded weekly, then r = 0.13/52 = 0.0025
Let's go ahead and substitute the values into the formula and solve for t;
[tex]\begin{gathered} 2P=P(1+0.0025)^t \\ \frac{2P}{P}=(1.0025)^t \\ 2=(1.0025)^t \end{gathered}[/tex]Let's now take the natural log of both sides;
[tex]\begin{gathered} \ln 2=\ln (1.0025)^t \\ \ln 2=t\cdot\ln (1.0025) \\ t=\frac{\ln 2}{\ln (1.0025)} \\ t=277.60\text{ w}eeks \\ t\approx\frac{2.77.60}{52}=5.34\text{ years} \end{gathered}[/tex]
