# Find Time In Continuous Compound Interest Word Problems Worksheet

## 6 problems

To find time in continuous compound interest word problems, use the formula $$A = Pe^{rt}$$ and solve for $$t$$. This involves taking the natural logarithm of both sides to isolate $$t$$, resulting in t = \frac{\ln(\frac{A}{P})}{r}. For detailed practice, refer to find time in continuous compound interest word problems worksheet which provides various examples and step-by-step solutions to enhance understanding.

Algebra 2
Exponential Functions

## How Will This Worksheet on 'Find Time in Continuous Compound Interest Word Problems' Benefit Your Student's Learning?

• Enhances students' ability to think critically and logically by manipulating and interpreting complex mathematical formulas.
• Helps students make informed financial decisions, such as planning for savings and investments, by understanding how to calculate the time required for investments to grow.
• Improves analytical skills by breaking down word problems into smaller, manageable parts and using the correct math formulas.
• Builds confidence in math skills, encouraging students to tackle more challenging concepts and problems.

## How to Find Time in Continuous Compound Interest Word Problems?

• Determine the final amount (A), the principal amount (P), and the annual interest rate (r) from the problem.
• Use the continuous compound interest formula $$A = Pe^{rt}$$.
• Divide both sides of the equation by $$P$$ to get \frac{A}{P} = e^{rt}.
• Take the natural logarithm of both sides and solve for $$t$$ using t = \frac{\ln(\frac{A}{P})}{r}.

## Solved Example

Q. Emily received a bonus of $\9,000$ from her company and wants to invest it in an account to save for a dream vacation. Her investment account has a $10\%$ interest rate compounded continuously. How long will it take for her money to grow to $\24,420$$?$ $\newline$Use the formula $A = Pe^{rt}$, where $A$ is the balance (final amount), $P$ is the principal (starting amount), $e$ is the base of natural logarithms ($\approx 2.71828$), $r$ is the interest rate expressed as a decimal, and $t$ is the time in years. $\newline$Round your answer to the nearest tenth.
Solution:
1. Identify values: Identify the values for $P$, $A$, $r$, and $t$.$\newline$ $P = 9000$$\newline$ $A = 24420$$\newline$ $r = 0.10$
2. Use formula: Use the formula $A = Pe^{rt}$.$\newline$ $24420 = 9000 \times e^{0.10 \times t}$
3. Divide to isolate: Divide both sides by $9000$ to isolate $e^{0.10 \times t}$.$\newline$ $\frac{24420}{9000} = e^{0.10 \times t}$$\newline$ $2.7133333 = e^{0.10 \times t}$
4. Take natural logarithm: Take the natural logarithm of both sides to solve for $t$.$\newline$ $\ln(2.7133333) = 0.10 \cdot t$
5. Calculate logarithm: Calculate the natural logarithm. $\ln(2.7133333) \approx 0.998$$\newline$ $0.998 = 0.10 \cdot t$
6. Solve for $t$: Solve for $t$ by dividing both sides by $0.10$.$\newline$ $t = \frac{0.998}{0.10}$$\newline$ $t \approx 9.98$
7. Round to nearest tenth: Round to the nearest tenth.$\newline$ $t \approx 10.0$$\newline$So, it will take approximately 10.0 years for Emily's money to grow.

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