8. You have $5000 toward the purchase of a boat that will cost $6000. How long will it take the $5000 to grow to $6000 if it is invested at 5% compounded monthly? Solve algebraically. Round to the nearest hundredth.
8. You have $5000 toward the purchase of a boat that will cost $6000. How long will it take the $5000 to grow to $6000 if it is invested at 5% compounded monthly? Solve algebraically. Round to the nearest hundredth.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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![**Problem 8: Investment Growth Calculation**
You have $5,000 toward the purchase of a boat that will cost $6,000. How long will it take the $5,000 to grow to $6,000 if it is invested at 5% compounded monthly? Solve algebraically. Round to the nearest hundredth.
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To solve this problem, you can use the formula for compound interest:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
Where:
- \( A \) is the future value of the investment/loan, including interest ($6,000 in this case).
- \( P \) is the principal investment amount ($5,000).
- \( r \) is the annual interest rate (decimal) (0.05 for 5%).
- \( n \) is the number of times that interest is compounded per unit year (12 for monthly).
- \( t \) is the time the money is invested for in years.
Substitute the known values into the formula:
\[ 6000 = 5000 \left(1 + \frac{0.05}{12}\right)^{12t} \]
To isolate \( t \), follow these steps:
1. Divide both sides by 5000:
\[ 1.2 = \left(1 + \frac{0.05}{12}\right)^{12t} \]
2. Take the natural logarithm of both sides:
\[ \ln(1.2) = \ln\left(\left(1 + \frac{0.05}{12}\right)^{12t}\right) \]
Using the logarithmic identity \(\ln(a^b) = b \ln(a)\):
\[ \ln(1.2) = 12t \cdot \ln\left(1 + \frac{0.05}{12}\right) \]
3. Solve for \( t \):
\[ t = \frac{\ln(1.2)}{12 \cdot \ln\left(1 + \frac{0.05}{12}\right)} \]
4. Calculate \( t \) using a calculator, and round to the nearest hundredth.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc9ef355a-fb22-427b-b06e-27c19f3d746c%2Fec79ab8c-3364-4ff5-8dfd-bc2dadbd8826%2Fw1c0kub_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 8: Investment Growth Calculation**
You have $5,000 toward the purchase of a boat that will cost $6,000. How long will it take the $5,000 to grow to $6,000 if it is invested at 5% compounded monthly? Solve algebraically. Round to the nearest hundredth.
---
To solve this problem, you can use the formula for compound interest:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
Where:
- \( A \) is the future value of the investment/loan, including interest ($6,000 in this case).
- \( P \) is the principal investment amount ($5,000).
- \( r \) is the annual interest rate (decimal) (0.05 for 5%).
- \( n \) is the number of times that interest is compounded per unit year (12 for monthly).
- \( t \) is the time the money is invested for in years.
Substitute the known values into the formula:
\[ 6000 = 5000 \left(1 + \frac{0.05}{12}\right)^{12t} \]
To isolate \( t \), follow these steps:
1. Divide both sides by 5000:
\[ 1.2 = \left(1 + \frac{0.05}{12}\right)^{12t} \]
2. Take the natural logarithm of both sides:
\[ \ln(1.2) = \ln\left(\left(1 + \frac{0.05}{12}\right)^{12t}\right) \]
Using the logarithmic identity \(\ln(a^b) = b \ln(a)\):
\[ \ln(1.2) = 12t \cdot \ln\left(1 + \frac{0.05}{12}\right) \]
3. Solve for \( t \):
\[ t = \frac{\ln(1.2)}{12 \cdot \ln\left(1 + \frac{0.05}{12}\right)} \]
4. Calculate \( t \) using a calculator, and round to the nearest hundredth.
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