Respuesta :
The time required to the nearest day to make the deposit of $1000 to grow to $1 million at 3% compounded continuously is 84044 days.
In the question ,
it is given that
the deposit ( principle amount P) = $1000
the final amount (A) = 1 million = $1000000
the rate of interest (r) = 3% = 0.03
let the time taken to make $1000 to 1 million be " t ".
The continuous compounding interest is given by the formula
[tex]A = P\times e^{r t}[/tex]
where A is the final amount
P is the principle amount
r is the rate of interest
t is the time taken
Substituting the values of A , P , r and t from above , we get
1000000 = 1000×[tex]e^{0.03*t}[/tex]
10000000/1000 = [tex]e^{0.03t}[/tex]
1000 = [tex]e^{0.03t}[/tex]
taking ln on both the sides
㏑(1000) = ㏑([tex]e^{0.03t}[/tex])
㏑(1000) = (0.03t)*㏑(e)
6.9077 = (0.03t)*1 .... as ln(e) = 1 and ln(1000)=6.9077
6.9077 = 0.03*t
t = 6.907755/0.03
t = 230.2585
So , time required is 230.2585 years .
to convert into days , we multiply by 365.
time ( in days) = 230.2585×365
= 84044.3525
≈ 84044 days
Therefore , the time required to make the deposit of $1000 to grow to $1 million at 3% compounded continuously is 84044 days .
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