### Problem Statement Find the time required for an investment of $3,000 to grow to $8,000 at an interest rate of 8% per year, compounded quarterly. ### Multiple Choice Options - A) 50 years - B) 3 years - C) 12 years - D) none of these ### Hand-Drawn Solution The image includes a partial view of a hand-drawn solution on a paper, which could not be entirely transcribed due to visibility issues. However, it appears that a formula involving compound interest (`S = P(1 + (r/n))^(nt)`) might be used to solve the problem. ### Explanation of the Required Formula To solve for the time required in compound interest problems, you can use the following formula: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] Where: - \( A \) is the amount of money accumulated after n years, including interest. - \( P \) is the principal amount (the initial sum of money, in this case, $3,000). - \( r \) is the annual interest rate (decimal). - \( n \) is the number of times that interest is compounded per year. - \( t \) is the time in years. **Steps to Find the Time Required:** \[ 8000 = 3000 \left(1 + \frac{0.08}{4}\right)^{4t} \] 1. Divide both sides of the equation by 3000: \[ \frac{8000}{3000} = \left(1 + \frac{0.08}{4}\right)^{4t} \] 2. Simplify: \[ \frac{8}{3} = \left(1 + 0.02\right)^{4t} \] \[ \frac{8}{3} = 1.02^{4t} \] 3. Take the natural logarithm (ln) of both sides: \[ \ln\left(\frac{8}{3}\right) = \ln\left(1.02^{4t}\right) \] 4. Simplify using the logarithm power rule: \[ \ln\left(\frac{8}{3}\right) = 4t \ln(1.02) \] 5. Solve for \( t \): \[
### Problem Statement Find the time required for an investment of $3,000 to grow to $8,000 at an interest rate of 8% per year, compounded quarterly. ### Multiple Choice Options - A) 50 years - B) 3 years - C) 12 years - D) none of these ### Hand-Drawn Solution The image includes a partial view of a hand-drawn solution on a paper, which could not be entirely transcribed due to visibility issues. However, it appears that a formula involving compound interest (`S = P(1 + (r/n))^(nt)`) might be used to solve the problem. ### Explanation of the Required Formula To solve for the time required in compound interest problems, you can use the following formula: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] Where: - \( A \) is the amount of money accumulated after n years, including interest. - \( P \) is the principal amount (the initial sum of money, in this case, $3,000). - \( r \) is the annual interest rate (decimal). - \( n \) is the number of times that interest is compounded per year. - \( t \) is the time in years. **Steps to Find the Time Required:** \[ 8000 = 3000 \left(1 + \frac{0.08}{4}\right)^{4t} \] 1. Divide both sides of the equation by 3000: \[ \frac{8000}{3000} = \left(1 + \frac{0.08}{4}\right)^{4t} \] 2. Simplify: \[ \frac{8}{3} = \left(1 + 0.02\right)^{4t} \] \[ \frac{8}{3} = 1.02^{4t} \] 3. Take the natural logarithm (ln) of both sides: \[ \ln\left(\frac{8}{3}\right) = \ln\left(1.02^{4t}\right) \] 4. Simplify using the logarithm power rule: \[ \ln\left(\frac{8}{3}\right) = 4t \ln(1.02) \] 5. Solve for \( t \): \[
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![### Problem Statement
Find the time required for an investment of $3,000 to grow to $8,000 at an interest rate of 8% per year, compounded quarterly.
### Multiple Choice Options
- A) 50 years
- B) 3 years
- C) 12 years
- D) none of these
### Hand-Drawn Solution
The image includes a partial view of a hand-drawn solution on a paper, which could not be entirely transcribed due to visibility issues. However, it appears that a formula involving compound interest (`S = P(1 + (r/n))^(nt)`) might be used to solve the problem.
### Explanation of the Required Formula
To solve for the time required in compound interest problems, you can use the following formula:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
Where:
- \( A \) is the amount of money accumulated after n years, including interest.
- \( P \) is the principal amount (the initial sum of money, in this case, $3,000).
- \( r \) is the annual interest rate (decimal).
- \( n \) is the number of times that interest is compounded per year.
- \( t \) is the time in years.
**Steps to Find the Time Required:**
\[ 8000 = 3000 \left(1 + \frac{0.08}{4}\right)^{4t} \]
1. Divide both sides of the equation by 3000:
\[ \frac{8000}{3000} = \left(1 + \frac{0.08}{4}\right)^{4t} \]
2. Simplify:
\[ \frac{8}{3} = \left(1 + 0.02\right)^{4t} \]
\[ \frac{8}{3} = 1.02^{4t} \]
3. Take the natural logarithm (ln) of both sides:
\[ \ln\left(\frac{8}{3}\right) = \ln\left(1.02^{4t}\right) \]
4. Simplify using the logarithm power rule:
\[ \ln\left(\frac{8}{3}\right) = 4t \ln(1.02) \]
5. Solve for \( t \):
\[](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5451aef3-1a55-4404-bb01-8f8933157334%2Fc2eea0d9-3aaa-4adb-83c9-30a3f1f287a9%2Fs58lzoc.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement
Find the time required for an investment of $3,000 to grow to $8,000 at an interest rate of 8% per year, compounded quarterly.
### Multiple Choice Options
- A) 50 years
- B) 3 years
- C) 12 years
- D) none of these
### Hand-Drawn Solution
The image includes a partial view of a hand-drawn solution on a paper, which could not be entirely transcribed due to visibility issues. However, it appears that a formula involving compound interest (`S = P(1 + (r/n))^(nt)`) might be used to solve the problem.
### Explanation of the Required Formula
To solve for the time required in compound interest problems, you can use the following formula:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
Where:
- \( A \) is the amount of money accumulated after n years, including interest.
- \( P \) is the principal amount (the initial sum of money, in this case, $3,000).
- \( r \) is the annual interest rate (decimal).
- \( n \) is the number of times that interest is compounded per year.
- \( t \) is the time in years.
**Steps to Find the Time Required:**
\[ 8000 = 3000 \left(1 + \frac{0.08}{4}\right)^{4t} \]
1. Divide both sides of the equation by 3000:
\[ \frac{8000}{3000} = \left(1 + \frac{0.08}{4}\right)^{4t} \]
2. Simplify:
\[ \frac{8}{3} = \left(1 + 0.02\right)^{4t} \]
\[ \frac{8}{3} = 1.02^{4t} \]
3. Take the natural logarithm (ln) of both sides:
\[ \ln\left(\frac{8}{3}\right) = \ln\left(1.02^{4t}\right) \]
4. Simplify using the logarithm power rule:
\[ \ln\left(\frac{8}{3}\right) = 4t \ln(1.02) \]
5. Solve for \( t \):
\[
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