Given the problem, we need to find the time (in years) it takes for an investment to grow from [tex]$5000 to $[/tex]6500 with a [tex]$6.25\%$[/tex] annual interest rate compounded quarterly.
Here's a detailed step-by-step solution:
### Step 1: Understand the Compound Interest Formula
The formula for compound interest is given by:
[tex]\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \][/tex]
- [tex]\(A\)[/tex] is the amount of money accumulated after [tex]\(t\)[/tex] years, including interest.
- [tex]\(P\)[/tex] is the principal amount (the initial amount of money).
- [tex]\(r\)[/tex] is the annual interest rate (in decimal form).
- [tex]\(n\)[/tex] is the number of times interest is compounded per year.
- [tex]\(t\)[/tex] is the number of years the money is invested for.
### Step 2: Define the Given Variables
From the problem, we know:
- [tex]\( P = 5000 \)[/tex] (the principal amount)
- [tex]\( A = 6500 \)[/tex] (the amount after interest)
- [tex]\( r = 6.25\% = 0.0625 \)[/tex] (the annual interest rate in decimal form)
- [tex]\( n = 4 \)[/tex] (because the interest is compounded quarterly)
### Step 3: Plug in the Known Values
Substitute the given values into the compound interest formula:
[tex]\[ 6500 = 5000 \left(1 + \frac{0.0625}{4}\right)^{4t} \][/tex]
### Step 4: Solve for the Compound Factor
Calculate the compound factor:
[tex]\[ 1 + \frac{r}{n} = 1 + \frac{0.0625}{4} = 1 + 0.015625 = 1.015625 \][/tex]
### Step 5: Isolate the Exponential Expression
Divide both sides by the principal amount:
[tex]\[ \frac{6500}{5000} = \left(1.015625\right)^{4t} \][/tex]
[tex]\[ 1.3 = (1.015625)^{4t} \][/tex]
### Step 6: Solve for the Time
To solve for [tex]\(t\)[/tex], we take the natural logarithm of both sides:
[tex]\[ \ln(1.3) = \ln((1.015625)^{4t}) \][/tex]
Utilize the properties of logarithms:
[tex]\[ \ln(1.3) = 4t \cdot \ln(1.015625) \][/tex]
[tex]\[ t = \frac{\ln(1.3)}{4 \cdot \ln(1.015625)} \][/tex]
### Step 7: Calculate the Time
Using the given results:
[tex]\[ \ln(1.3) \approx 0.26236426446749106 \][/tex]
[tex]\[ \ln(1.015625) \approx 0.015524247675986096 \][/tex]
[tex]\[
t = \frac{0.26236426446749106}{4 \cdot 0.015524247675986096} \approx 4.230539020200792
\][/tex]
### Step 8: Round to the Nearest Tenth
Finally, round the result to the nearest tenth:
[tex]\[ t \approx 4.2 \][/tex]
Thus, to the nearest tenth of a year, the person must leave the money in the bank for approximately [tex]\(4.2\)[/tex] years.
