(2022 Spring Final Q2) The equations below show Calvin’s estimated per-month benefit of watching n movies on an online movie platform. (Assume that n can take only integer values.)

--------------------------------------------------------------------------------
Total benefit (dollars): TB(n) = 653n − 7n2
Marginal benefit (dollars): MB(n) = 660 − 14n
Average benefit (dollars): AB(n) = 653 − 7n
--------------------------------------------------------------------------------

Upon paying a monthly subscription fee of 24 dollars, Calvin will be eligible to watch any movie at P dollars per movie. If Calvin decides to watch 27 movies per month under the scheme, then we can infer that the price per movie (P) is less than or equal to Answer field 1
 dollars and more than Answer field 2
 dollars.

We are given the total benefit function \( TB(n) = 653n - 7n^2 \), marginal benefit function \( MB(n) = 660 - 14n \), and average benefit function \( AB(n) = 653 - 7n \).

Additionally, Calvin will pay a monthly subscription fee of 24 dollars and a price per movie \( P \) dollars. He decides to watch 27 movies per month, and we are asked to determine the bounds for \( P \) (i.e., it is less than or equal to some value and more than some value).

### Step 1: Calculate Total Benefit for 27 movies
First, let's calculate the total benefit for watching 27 movies per month using the total benefit function \( TB(n) \):
\[
TB(27) = 653(27) - 7(27)^2
\]
\[
TB(27) = 17631 - 7(729)
\]
\[
TB(27) = 17631 - 5103 = 12528 \text{ dollars.}
\]

### Step 2: Calculate Marginal Benefit for 27 movies
Now, let's calculate the marginal benefit for watching the 27th movie using the marginal benefit function \( MB(n) \):
\[
MB(27) = 660 - 14(27)
\]
\[
MB(27) = 660 - 378 = 282 \text{ dollars.}
\]
This tells us that the additional benefit Calvin gets from watching the 27th movie is 282 dollars.

### Step 3: Set up the inequality for price per movie \( P \)
For Calvin to watch 27 movies, the marginal benefit of the 27th movie must be greater than or equal to the price he pays per movie (i.e., \( MB(27) \geq P \)). Thus:
\[
P \leq 282 \text{ dollars.}
\]
So the price per movie \( P \) must be less than or equal to 282 dollars.

### Step 4: Consider the total cost
Calvin's total cost for watching 27 movies includes the 24-dollar subscription fee plus \( P \times 27 \) for the price of each movie. The total cost must be less than or equal to the total benefit from watching 27 movies (since Calvin decides to watch 27 movies):
\[
24 + 27P \leq 12528
\]
Solving for \( P \):
\[
27P \leq 12528 - 24
\]
\[
27P \leq 12504
\]
\[
P \geq \frac{12504}{27} \approx 463 \text{ dollars.}
\]

### Conclusion:
Thus, the price per movie \( P \) is **less than or equal to 282 dollars** (based on marginal benefit) and **more than approximately 463 dollars** (based on total benefit). Therefore, the answers for the price per movie are:
- **Answer field 1** (less than or equal to) = 282 dollars.
- **Answer field 2** (more than) = approximately 463 dollars.

Do you want more details on any of these steps, or do you have any questions?

Here are 5 related questions:
1. How would the answer change if Calvin watches a different number of movies, say 30 movies per month?
2. How does the marginal benefit function influence Calvin's decision on how many movies to watch?
3. Can we solve this problem using calculus? If so, what would the critical points indicate?
4. What is the interpretation of the average benefit function in this context?
5. How does the subscription fee affect the calculation of the optimal number of movies Calvin should watch?

**Tip:** Marginal benefit is the additional benefit from consuming one more unit of a good, and it helps determine the optimal quantity to consume when comparing with the cost.

(2022 Spring Final Q2) The equations below show Calvin’s estimated per-month benefit of watching n movies on an online movie platform. (Assume that n can take only integer values.)

--------------------------------------------------------------------------------
Total benefit (dollars): TB(n) = 653n − 7n^2
Marginal benefit (dollars): MB(n) = 660 − 14n
Average benefit (dollars): AB(n) = 653 − 7n
--------------------------------------------------------------------------------

Upon paying a monthly subscription fee of 24 dollars, Calvin will be eligible to watch any movie at P dollars per movie. If Calvin decides to watch 27 movies per month under the scheme, then we can infer that the price per movie (P) is less than or equal to Answer field 1 dollars and more than Answer field 2 dollars.

Explore a detailed step-by-step solution to determine the price bounds for movie tickets using total and marginal benefit functions. This question involves calculating the total benefit for watching 27 movies, analyzing marginal benefits, and setting up inequalities to infer price per movie. Suitable for economics and mathematics students at the college level.

Math Problem Statement

Total benefit (dollars): TB(n) = 653n − 7n2 Marginal benefit (dollars): MB(n) = 660 − 14n Average benefit (dollars): AB(n) = 653 − 7n

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