Respuesta :
Using compound interest, it is found that the approximate difference in the number of years that calvin and makayla have their money invested is given as follows:
Makayla invests her money 2 years longer.
What is compound interest?
The amount of money earned, in compound interest, after t years, is given by:
[tex]A(t) = P\left(1 + \frac{r}{n}\right)^{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.
For Calvin, the parameters are given as follows:
[tex]P = 400, r = 0.05, n = 12, A(t) = 658.8[/tex]
Hence:
[tex]A(t) = P\left(1 + \frac{r}{n}\right)^{nt}[/tex]
[tex]658.8= 400\left(1 + \frac{0.05}{12}\right)^{12t}[/tex]
[tex]\left(1 + \frac{0.05}{12}\right)^{12t} = \frac{658.8}{400}[/tex]
[tex](1.00416666667)^{12t} = 1.647[/tex]
[tex]\log{(1.00416666667)^{12t}} = \log{1.647}[/tex]
[tex]12t\log{1.00416666667} = \log{1.647}[/tex]
[tex]t = \frac{\log{1.647}}{12\log{1.00416666667}}[/tex]
[tex]t = 10[/tex]
For Makayla, the parameters are given as follows:
[tex]P = 300, r = 0.06, n = 4, (t) = 613.4[/tex]
Hence:
[tex]A(t) = P\left(1 + \frac{r}{n}\right)^{nt}[/tex]
[tex]613.4 = 300\left(1 + \frac{0.06}{4}\right)^{4t}[/tex]
[tex]\left(1 + \frac{0.06}{4}\right)^{4t} = \frac{613.4}{300}[/tex]
[tex](1.015)^{4t} = 2.0447[/tex]
[tex]\log{(1.015)^{12t}} = \log{2.0447}[/tex]
[tex]12t\log{1.015} = \log{2.0447}[/tex]
[tex]t = \frac{\log{2.0447}}{4\log{1.015}}[/tex]
[tex]t = 12[/tex]
12 - 10 = 2, hence Makayla invests her money 2 years longer.
More can be learned about compound interest at https://brainly.com/question/25781328
