Respuesta :
First compute how many years are there from 1626:
2014-1626= 388 years.
Let the annual rate be r. Then the formula is the following:
[tex]24(1+r)^{388}=6,000,000,000[/tex]
We solve the above equation in order to find the value of r like this:
[tex](1+r)^{388}=\frac{6,000,000,000}{24}\\ 388 log(1+r)=\log(\frac{6,000,000,000}{24})\\ \log(1+r)=\log(\frac{6,000,000,000}{24})\times\frac{1}{388}\\
1+r=\exp(\log(\frac{6,000,000,000}{24})\times\frac{1}{388})[/tex]
Using a calculator we get:
r=0.05
The annual rate is then 5%
2014-1626= 388 years.
Let the annual rate be r. Then the formula is the following:
[tex]24(1+r)^{388}=6,000,000,000[/tex]
We solve the above equation in order to find the value of r like this:
[tex](1+r)^{388}=\frac{6,000,000,000}{24}\\ 388 log(1+r)=\log(\frac{6,000,000,000}{24})\\ \log(1+r)=\log(\frac{6,000,000,000}{24})\times\frac{1}{388}\\
1+r=\exp(\log(\frac{6,000,000,000}{24})\times\frac{1}{388})[/tex]
Using a calculator we get:
r=0.05
The annual rate is then 5%
The principal is
P = $24
Calculate the duration.
t = 2014 - 1626 = 388 years
The value after 388 years is
A = $6 x 10⁹
For continuous compounding, the compounding interval is
n = 365
Let r = the rate.
Then use the formula
[tex]P(1 + \frac{r}{n} )^{nt} = A[/tex]
That is,[tex]24(1+ \frac{r}{365} )^{365*388} = 6 \times 10^{9}\\(1+ \frac{r}{365})^{ 141620} = \frac{6 \times 10^{9}}{24}= 2.5 \times 10^{8} \\1 + \frac{r}{365} =(2.5 \times 10^{8})^{1/141620} = 1.00013655[/tex]
Hence obtain
r/365 = 1.00013655 - 1 = 0.00013655
r = (0.00013655)*(365) = 0.0498 = 4.98%
Answer: 5.0% (to 1 decimal place)
P = $24
Calculate the duration.
t = 2014 - 1626 = 388 years
The value after 388 years is
A = $6 x 10⁹
For continuous compounding, the compounding interval is
n = 365
Let r = the rate.
Then use the formula
[tex]P(1 + \frac{r}{n} )^{nt} = A[/tex]
That is,[tex]24(1+ \frac{r}{365} )^{365*388} = 6 \times 10^{9}\\(1+ \frac{r}{365})^{ 141620} = \frac{6 \times 10^{9}}{24}= 2.5 \times 10^{8} \\1 + \frac{r}{365} =(2.5 \times 10^{8})^{1/141620} = 1.00013655[/tex]
Hence obtain
r/365 = 1.00013655 - 1 = 0.00013655
r = (0.00013655)*(365) = 0.0498 = 4.98%
Answer: 5.0% (to 1 decimal place)
