An investor is presented with a choice of two​ investments: an established furniture store and a new computer 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 furniture store is f left parenthesis t right parenthesis equals 18 comma 000​, and the rate of flow of income from the computer store is expected to be g left parenthesis t right parenthesis equals 16 comma 000 e Superscript 0.04 t. Compare the future values of these investments to determine which is the better choice over the next 6 years.
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Part 1
The future value of the furniture store is ​$
  
129999. 
​(Round to the nearest dollar as​ needed.) 
Part 2
The future value of the computer 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 6 years, we need to calculate the future value (FV) of both the furniture store and the computer store using the given income streams and the fact that the income is compounded continuously at a rate of 6%.

### Part 1: Furniture Store
The rate of flow of income from the furniture store is given by:
\[ f(t) = 18,000 \]

Since the income stream is continuous and the interest rate is compounded continuously, the future value for the furniture store over $ t $ years is calculated using the formula:
\[ \text{FV}_{\text{furniture}} = \int_0^6 f(t) \cdot e^{0.06(6 - t)} \, dt \]

Given that $ f(t) = 18,000 $, the future value calculation simplifies to:
\[ \text{FV}_{\text{furniture}} = 18,000 \cdot \int_0^6 e^{0.06(6 - t)} \, dt \]

This calculation results in the future value of the furniture store being approximately \$129,999.

### Part 2: Computer Store
The rate of flow of income from the computer store is given by:
\[ g(t) = 16,000 e^{0.04t} \]

The future value for the computer store over $ t $ years is calculated using the same type of formula:
\[ \text{FV}_{\text{computer}} = \int_0^6 g(t) \cdot e^{0.06(6 - t)} \, dt \]

Substituting the given $ g(t) $ into the formula:
\[ \text{FV}_{\text{computer}} = \int_0^6 16,000 e^{0.04t} \cdot e^{0.06(6 - t)} \, dt \]

Simplifying further:
\[ \text{FV}_{\text{computer}} = 16,000 \cdot \int_0^6 e^{0.04t + 0.36 - 0.06t} \, dt \]
\[ \text{FV}_{\text{computer}} = 16,000 \cdot e^{0.36} \cdot \int_0^6 e^{-0.02t} \, dt \]

This integral evaluates to:
\[ \text{FV}_{\text{computer}} = 16,000 \cdot e^{0.36} \cdot \left[ \frac{e^{-0.02t}}{-0.02} \right]_0^6 \]

After performing the calculations:
\[ \text{FV}_{\text{computer}} \approx \$138,368 \]

So, the future value of the computer store investment is approximately **\$138,368**.

### Conclusion:
The computer store has a higher future value over the next 6 years compared to the furniture store, making it the better investment choice based on these calculations.

Would you like more details or have any questions?

### Relative Questions:
1. How does continuous compounding affect future value calculations?
2. What is the significance of the rate of flow of income in investment calculations?
3. How does the choice of investment time horizon influence the better investment option?
4. What would be the future value if the interest rate was compounded annually instead of continuously?
5. Can the future value of the computer store change with different growth rates in the rate of flow of income?
6. How would an increase in the interest rate impact the future value of both investments?
7. What is the role of the exponential function in the future value calculation of the computer store?
8. How would the future values compare if the investment period was extended to 10 years?

### Tip:
When comparing investments, consider both the rate of return and the growth potential of the income stream over time to make the most informed decision.

Compare the future values of investments in an established furniture store and a new computer store over 6 years. Detailed calculations using continuous compounding, income streams, and exponential functions. Determine which investment yields higher returns.

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