Unitary Method
The word “unitary” comes from the word “unit”, which means a single and complete entity. In this method, we find the value of a unit product from the given number of products, and then we solve for the other number of products.
Speed, Time, and Distance
Imagine you and 3 of your friends are planning to go to the playground at 6 in the evening. Your house is one mile away from the playground and one of your friends named Jim must start at 5 pm to reach the playground by walk. The other two friends are 3 miles away.
Profit and Loss
The amount earned or lost on the sale of one or more items is referred to as the profit or loss on that item.
Units and Measurements
Measurements and comparisons are the foundation of science and engineering. We, therefore, need rules that tell us how things are measured and compared. For these measurements and comparisons, we perform certain experiments, and we will need the experiments to set up the devices.
![**Question 7:** At what interest rate compounded continuously must money be invested in order to triple in 10 years?
**Context:** This question is focused on continuously compounded interest and requires understanding of the exponential growth formula in the context of financial mathematics.
**Explanation:**
When dealing with continuously compounded interest, the formula to use is:
\[ A = P \cdot e^{(rt)} \]
Where:
- \( A \) is the amount of money accumulated after time \( t \).
- \( P \) is the principal amount (initial investment).
- \( r \) is the annual interest rate (as a decimal).
- \( t \) is the time the money is invested for (in years).
- \( e \) is the base of the natural logarithm (approximately equal to 2.71828).
**Problem Setup:**
- To triple the principal amount, we set \( A = 3P \).
- The time period \( t \) is given as 10 years.
- We must find the interest rate \( r \).
**Solution:**
1. Substitute the values into the formula:
\[ 3P = P \cdot e^{(10r)} \]
2. Divide both sides by \( P \):
\[ 3 = e^{(10r)} \]
3. Take the natural logarithm of both sides to solve for \( r \):
\[ \ln(3) = 10r \]
4. Divide both sides by 10:
\[ r = \frac{\ln(3)}{10} \]
**Conclusion:**
The interest rate \( r \) required for the money to triple in 10 years, when compounded continuously, is:
\[ r = \frac{\ln(3)}{10} \approx 0.10986 \text{ or } 10.986\% \]
Understanding this problem involves a good grasp of logarithms and exponential functions, particularly in the context of financial growth.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb3c4489a-e4bd-4ff1-89b5-d5833f06911e%2F99daf9b9-0b9f-421c-8a22-41ab404b6021%2Focy1lb_processed.png&w=3840&q=75)
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