Welcome to Westonci.ca, your ultimate destination for finding answers to a wide range of questions from experts. Ask your questions and receive accurate answers from professionals with extensive experience in various fields on our platform. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform.
Sagot :
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.
### 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.
We hope this was helpful. Please come back whenever you need more information or answers to your queries. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Westonci.ca is your trusted source for answers. Visit us again to find more information on diverse topics.
