Respuesta :
Answer:
The time required for this is 7.61 years.
Step-by-step explanation:
The compound interest formula is given by:
[tex]A(t) = P(1 + \frac{r}{n})^{nt}[/tex]
Where 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 unit year and t is the time in years for which the money is invested or borrowed.
In this exercise, we have:
To find t for which [tex]A(t) = 8800[/tex] when [tex]P = 5000, r = 0.075, n = 4[/tex],
Then
[tex]A(t) = P(1 + \frac{r}{n})^{nt}[/tex]
[tex]8800 = 5000(1 + \frac{0.075}{4})^{4t}[/tex]
[tex](1.01875)^{4t} = \frac{8800}{5000}[/tex]
[tex](1.01875)^{4t} = 1.76[/tex]
We have the following logarithm property:
[tex]\log{a^{t}} = t\log{a}[/tex]
Then
[tex]\log{(1.01875)^{4t}} = \log{1.76}[/tex]
[tex]4t\log{1.01875} = \log{1.76}[/tex]
[tex]t = \frac{\log{1.76}}{4\log{1.01875}}[/tex]
[tex]t = 7.61[/tex]
The time required for this is 7.61 years.
The time required for an investment of 5000 dollars to grow to 8800 dollars is 7.61 years and this can be determined by using the compound interest formula.
Given :
An investment of 5000 dollars to grow to 8800 dollars at an interest rate of 7.5 percent per year.
The formula of compound interest is used to determine the time required for an investment of 5000 dollars to grow to 8800 dollars.
The compound interest formula is given by:
[tex]\rm A(t) = P(1+\dfrac{r}{n})^{nt}[/tex]
where A is the final amount, P is the principal amount, r is the interest rate, n is the number of times that interest is compounded per unit year and t is the time required.
Now, put the known terms in the above equation.
[tex]\rm 8800 = 5000\left(1+\dfrac{0.075}{4}\right)^{4t}[/tex]
[tex](1.01875)^{4t} = \dfrac{8800}{5000}[/tex]
[tex](1.01875)^{4t} = 1.76[/tex]
Now, take the log on both sides in the above equation.
[tex]\rm (4t)log(1.01875) = log(1.76)[/tex]
[tex]\rm t = \dfrac{log(1.76)}{4log(1.01875)}[/tex]
t = 7.61
The time required is 7.61 years.
For more information, refer to the link given below:
https://brainly.com/question/22803385
