Questions: The current revenue for a company is 6 million. The chief officer (CFO) of the company estimates that over the next 5 years, revenues will increase at a continuous rate of between 5% and 6% each year. The function that models the estimated 5% increase is (R5%=6 e^0.05 t) and the function that models the estimated 6% increase is (R6%=6 e^0.06 t). For each model, (t) represents the number of years from now. Approximate the total difference in revenue under the two models. Answer to the nearest dollar.

Solution Steps

To find the total difference in revenue under the two models over the next 5 years, we need to calculate the revenue for each model at \( t = 5 \) years and then find the difference between these two values. The revenue for each model is given by the exponential functions \( R_{5\%} = 6e^{0.05 \times 5} \) and \( R_{6\%} = 6e^{0.06 \times 5} \). Finally, subtract the revenue from the 5% model from the 6% model to find the difference.

The revenue model for a 5% increase over 5 years is given by:

\[ R_{5\%} = 6 e^{0.05 \cdot 5} \]

Calculating this, we find:

\[ R_{5\%} \approx 7.7042 \text{ million dollars} \]

The revenue model for a 6% increase over 5 years is given by:

\[ R_{6\%} = 6 e^{0.06 \cdot 5} \]

Calculating this, we find:

\[ R_{6\%} \approx 8.0992 \text{ million dollars} \]

To find the total difference in revenue between the two models, we subtract the revenue from the 5% model from the revenue from the 6% model:

\[ \text{Difference} = R_{6\%} - R_{5\%} \approx 8.0992 - 7.7042 = 0.3950 \text{ million dollars} \]

Converting this difference to dollars gives:

\[ \text{Difference in dollars} \approx 395000 \]

Final Answer