We are tasked with finding the approximate monthly sales of Mona Lisa paint-by-number sets when the retail price $p$ is set at the optimal price determined in part (b), which is 77 hundred yen, or ¥7,700.

The demand function is given as:

$q = 104e^p - \frac{3p^2}{2}$

Where:

• $q$ is the demand in monthly sales.
• $p$ is the price in hundreds of yen.

Step-by-step solution:

1. Substitute $p = 77$ (since the optimal price in part (b) is 77 hundred yen) into the demand formula:

$q = 104e^{77} - \frac{3(77)^2}{2}$

2. Calculate the two components of the formula:
   • First, $104e^{77}$ is an exponential term.
   • Second, $\frac{3(77)^2}{2}$ involves squaring 77, multiplying by 3, and dividing by 2.

Key Steps:

• Simplify the exponential term.
• Solve the polynomial term $\frac{3(77)^2}{2}$.

Let me calculate these values for you.The exponential term in the demand function leads to an extremely large value, indicating that the exponential growth component $104e^{77}$ is overwhelmingly large, producing a result on the order of $10^{35}$. This suggests that the model used here may not be practically applicable at such a high price point due to the steep growth in demand predicted by the exponential function.

If you are interested in a more reasonable model, there may be an error or a more realistic constraint to place on the parameters. Would you like further clarification or adjustments on this?

Expanding your understanding:

1. Why does the exponential function grow so rapidly?
2. How can exponential growth affect demand in real-world scenarios?
3. What kind of price-demand models work better for real-life products?
4. What does it mean when a mathematical model yields impractical results?
5. How can demand functions be adjusted for more realistic predictions?

Tip:

Exponential terms in demand models can sometimes indicate extreme sensitivity, so they need to be carefully checked to ensure they fit the context of real-world behaviors.