Let's solve this compound interest problem step-by-step to find out how long it will take for an investment of [tex]$4000 to grow to $[/tex]20,000 with an annual interest rate of 5.2% compounded daily.
### Step 1: Understand the Formula
The formula for compound interest is:
[tex]\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \][/tex]
where:
- [tex]\( A \)[/tex] is the final amount (in this case, [tex]$20,000)
- \( P \) is the principal amount (initial deposit, in this case, $[/tex]4000)
- [tex]\( r \)[/tex] is the annual interest rate (in decimal form, so 5.2% becomes 0.052)
- [tex]\( n \)[/tex] is the number of times the interest is compounded per year (for daily compounding, this is 365)
- [tex]\( t \)[/tex] is the time in years that we want to find
### Step 2: Rearrange the Formula
We need to solve for [tex]\( t \)[/tex]. Rearranging the formula to isolate [tex]\( t \)[/tex], we get:
[tex]\[ t = \frac{\log(\frac{A}{P})}{n \cdot \log(1 + \frac{r}{n})} \][/tex]
### Step 3: Plug in the Values
Let’s substitute the values into the formula:
- [tex]\( A = 20000 \)[/tex]
- [tex]\( P = 4000 \)[/tex]
- [tex]\( r = 0.052 \)[/tex]
- [tex]\( n = 365 \)[/tex]
[tex]\[ t = \frac{\log(\frac{20000}{4000})}{365 \cdot \log(1 + \frac{0.052}{365})} \][/tex]
### Step 4: Simplify the Formula
Calculate each part of the formula:
- [tex]\(\frac{20000}{4000} = 5\)[/tex]
- [tex]\(\log(5)\)[/tex] (let's approximate this using a logarithm calculator)
- [tex]\(1 + \frac{0.052}{365}\)[/tex] (calculate this first)
[tex]\[ 1 + \frac{0.052}{365} \approx 1.00014246575 \][/tex]
- Now find the log of the above value (using a logarithm calculator again)
### Step 5: Solve for [tex]\( t \)[/tex]
After plugging these values and solving the logarithms, you will find:
[tex]\[ t \approx 30.952933742411332 \][/tex]
### Step 6: Convert to Years and Days
For the approximate number of years:
- The integer part of [tex]\( t \)[/tex] is the number of full years, which is [tex]\( 30 \)[/tex] years.
For the remaining days:
- Calculate the fractional part: [tex]\( 0.952933742411332 \)[/tex]. Multiply this by 365 to get the remaining days:
[tex]\[ 0.952933742411332 \times 365 \approx 347.8208159801363 \][/tex]
So, for extra credit:
- In exact terms, this is approximately [tex]\( 30 \)[/tex] years and [tex]\( 348 \)[/tex] days (rounding 347.8208159801363 to the nearest whole number).
### Final Answer:
Approximation:
- It will take approximately between 30 to 31 years for your investment to be worth [tex]$20,000.
Exact Answer for Extra Credit:
- It will take approximately 30 years and 348 days for your investment to grow to $[/tex]20,000. The most exact representation of the time needed is 30.952933742411332 years or 30 years and 348 days.
