The estimated monthly sales of Mona Lisa paint-by-number sets is given by the formula 
q = 104ep − 3p2⁄2,
 where q is the demand in monthly sales and p is the retail price in hundreds of yen.
(a)
Determine the price elasticity of demand E when the retail price is set at ¥400.
E = 
44
 
Correct: Your answer is correct.
Interpret your answer.
The demand is going 
down
 
Correct: Your answer is correct.
 by 
44
 
Correct: Your answer is correct.
 % per 1% increase in price at that price level. Thus, a large price 
decrease
 
Correct: Your answer is correct.
 is advised.
(b)
At what price will revenue be a maximum? (Round your answer to the nearest integer.)
77
 
Correct: Your answer is correct.
 yen
(c)
Approximately how many paint-by-number sets will be sold per month at the price in part (b)? (Round your answer to the nearest integer.)
find
 paint-by-number sets per month

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.

The estimated monthly sales of Mona Lisa paint-by-number sets is given by the formula q = 104ep − 3p²/2, where q is the demand in monthly sales and p is the retail price in hundreds of yen. (a) Determine the price elasticity of demand E when the retail price is set at ¥400. E = 44. Interpret your answer. The demand is going down by 44% per 1% increase in price at that price level. Thus, a large price decrease is advised. (b) At what price will revenue be a maximum? 77 yen. (c) Approximately how many paint-by-number sets will be sold per month at the price in part (b)?

Explore the price elasticity of demand and revenue maximization for Mona Lisa paint-by-number sets, using advanced economic formulas. Understand how a 44% decrease in demand occurs per 1% price increase, and learn the optimal price for maximum revenue. Suitable for college-level students studying economics or applied mathematics.

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