Question content area top
Part 1
An investor is presented with a choice of two​ investments: an established clothing store and a new book store. Each choice requires the same initial investment and each produces a continuous income stream of 6​%, compounded continuously. The rate of flow of income from the clothing store is f left parenthesis t right parenthesis equals 18 comma 000​, and the rate of flow of income from the book store is expected to be g left parenthesis t right parenthesis equals 17 comma 000 e Superscript 0.04 t. Compare the future values of these investments to determine which is the better choice over the next 8 years.
Question content area bottom
Part 1
The future value of the clothing store is ​$
  
enter your response here. 
​(Round to the nearest dollar as​ needed.) 

To compare the future values of the two investments over the next 8 years, we need to calculate the future value (FV) of the continuous income streams for both the clothing store and the book store.

### 1. Future Value of the Clothing Store Investment:
The rate of flow of income from the clothing store is constant: $ f(t) = 18,000 $.

The future value of a continuous income stream is given by the formula:
\[
FV = \int_0^T f(t) e^{r(T-t)} \, dt
\]
where:
- $ f(t) $ is the rate of income flow,
- $ r $ is the continuous compounding rate (6%, or 0.06),
- $ T $ is the time period (8 years).

Since the income rate $ f(t) $ is constant ($ 18,000 $), we can simplify the formula:
\[
FV = 18,000 \times \int_0^8 e^{0.06(8-t)} \, dt
\]

Let's compute this integral.

### 2. Future Value of the Book Store Investment:
The rate of flow of income from the book store is $ g(t) = 17,000 e^{0.04t} $.

Similarly, the future value for the book store is:
\[
FV = \int_0^8 17,000 e^{0.04t} e^{0.06(8-t)} \, dt
\]
Simplifying:
\[
FV = 17,000 \times \int_0^8 e^{0.04t + 0.06(8-t)} \, dt
\]
\[
= 17,000 \times \int_0^8 e^{0.04t + 0.48 - 0.06t} \, dt
\]
\[
= 17,000 \times e^{0.48} \times \int_0^8 e^{-0.02t} \, dt
\]

Let's calculate both integrals to compare the future values. 

### Calculation:

We'll compute the future value of both investments over the 8-year period.

#### For the clothing store:
\[
FV_{\text{clothing}} = 18,000 \times \int_0^8 e^{0.06(8-t)} \, dt
\]

#### For the book store:
\[
FV_{\text{book}} = 17,000 \times e^{0.48} \times \int_0^8 e^{-0.02t} \, dt
\]

I'll calculate these integrals to find the precise future values.### Results:
- **Future Value of the Clothing Store Investment**: \$184,822
- **Future Value of the Book Store Investment**: \$203,105

### Conclusion:
The future value of the book store investment is higher over the next 8 years compared to the clothing store. Therefore, the book store is the better choice based on the future value of the income streams.

Would you like more details, or do you have any questions?

### Related Questions:
1. How does continuous compounding affect investment returns?
2. What would be the future value if the investment period was 10 years instead of 8?
3. How does the initial rate of income flow influence the future value?
4. What happens to the future value if the interest rate is reduced to 4%?
5. Can we calculate the present value of these income streams?
6. What if the rate of flow for the book store changes to $ g(t) = 17,000e^{0.05t} $?
7. How would the comparison change if the compounding was done annually instead of continuously?
8. What are the risks associated with selecting the book store based on these calculations?

### Tip:
Understanding how to calculate future values with continuous compounding is crucial for evaluating long-term investments and comparing different financial opportunities.

Compare the future values of two investments, an established clothing store with a constant income stream and a new book store with income increasing exponentially. Calculate future values using continuous compounding and integral calculus. Detailed solution and comparison provided.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation