Respuesta :
Using compound interest, it is found that it will take 13.5 years.
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The compound interest formula is given by:
[tex]A(t) = P(1 + \frac{r}{n})^{nt}[/tex]
In which:
- A(t) is the amount of money after t years.
- P is the principal(the initial sum of money).
- r is the interest rate(as a decimal value).
- n is the number of times that interest is compounded per year.
- t is the time in years for which the money is invested or borrowed.
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- Investment of $1000, thus [tex]P = 1000[/tex]
- Interest rate of 3%, thus [tex]r = 0.03[/tex]
- Compounded monthly, thus [tex]n = 12[/tex]
- Grow to $1500, thus [tex]A(t) = 1500[/tex].
We have to solve for t, thus:
[tex]A(t) = P(1 + \frac{r}{n})^{nt}[/tex]
[tex]1500 = 1000(1 + \frac{0.03}{12})^{12t}[/tex]
[tex](1.0025)^{12t} = \frac{1500}{1000}[/tex]
[tex](1.0025)^{12t} = 1.5[/tex]
[tex]\log{(1.0025)^{12t}} = \log{1.5}[/tex]
[tex]12t\log{1.0025} = \log{1.5}[/tex]
[tex]12t = \frac{\log{1.5}}{\log{1.0025}}[/tex]
[tex]12t = 162.388691431 [/tex]
[tex]t = \frac{162.388691431}{12}[/tex]
[tex]t = 13.5[/tex]
It will take 13.5 years.
A similar problem is given at https://brainly.com/question/23781391
