Answer :
Answer:
The rate is needed is 1.037%.
Step-by-step explanation:
Given : Suppose $1,500 is compounded weekly for 46 years. If no other deposits are made.
To find : What rate is needed for the balance to triple in that time?
Solution :
Applying compound interest formula,
[tex]A=P(1+\frac{r}{n})^{nt}[/tex]
Where, P is the principal
A is the amount
The balance to triple in that time i.e. A=3P
r is the rate
t is the time t=46 years
Compounded weekly so n=52
Substitute the value in the formula,
[tex]3P=P(1+\frac{r}{52})^{52\times 46}[/tex]
[tex]3=(1+\frac{r}{52})^{2392}[/tex]
Taking log both side,
[tex]\log 3=2392\ log(1+\frac{r}{52})[/tex]
[tex]\frac{\log 3}{2392}=\ log(1+\frac{r}{52})[/tex]
[tex]0.00019946=\ log(1+\frac{r}{52})[/tex]
Taking exponential both side,
[tex]e^{0.00019946}=1+\frac{r}{52}[/tex]
[tex]1.000199-1=\frac{r}{52}[/tex]
[tex]0.000199=\frac{r}{52}[/tex]
[tex]r=0.000199\times 52[/tex]
[tex]r=0.010372[/tex]
Into percentage,
[tex]r=0.010372\times 100[/tex]
[tex]r=1.0372[/tex]
Therefore, the rate is needed is 1.037%.