**(2021 Summer Midterm)** The equation below shows Ben’s benefit of consuming n cups of tea and tea is perfectly divisible and can be consumed at any arbitrary quantity (say, 3.14 cups).

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Total benefit: TB(n) = 229n − 10n2

Average benefit: AB(n) = 229 − 10n

Marginal benefit: MB(n) = 229 − 20n

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Suppose the price is HKD 33 per cup. Ben’s economic surplus from consuming tea would be HKD [**Answer**].

(In decimal numbers, with two decimal places, please.)

To calculate Ben's economic surplus, we need to determine the number of cups of tea Ben will consume and the associated surplus.

### Steps:

1. **Determine Optimal Consumption (n):**
   Ben will consume tea up to the point where the marginal benefit (MB) equals the price of the tea (P).

   \[
   MB(n) = P
   \]

   Given:
   - Marginal benefit: \( MB(n) = 229 - 20n \)
   - Price: \( P = 33 \)

   Setting MB equal to the price:

   \[
   229 - 20n = 33
   \]

   Solving for \( n \):

   \[
   229 - 33 = 20n
   \]

   \[
   196 = 20n
   \]

   \[
   n = \frac{196}{20} = 9.8
   \]

   So, Ben will consume **9.8 cups** of tea.

2. **Calculate Total Benefit (TB):**
   Now we calculate Ben's total benefit from consuming \( n = 9.8 \) cups of tea.

   \[
   TB(9.8) = 229(9.8) - 10(9.8)^2
   \]

   First, calculate \( 229 \times 9.8 \):

   \[
   229 \times 9.8 = 2244.2
   \]

   Next, calculate \( 10 \times (9.8)^2 \):

   \[
   9.8^2 = 96.04
   \]

   \[
   10 \times 96.04 = 960.4
   \]

   Now, subtract:

   \[
   TB(9.8) = 2244.2 - 960.4 = 1283.8
   \]

3. **Calculate Total Cost (TC):**
   The total cost is simply the price per cup multiplied by the number of cups consumed:

   \[
   TC = 33 \times 9.8 = 323.4
   \]

4. **Calculate Economic Surplus (ES):**
   Economic surplus is the total benefit minus the total cost:

   \[
   ES = TB(n) - TC
   \]

   Substituting the values:

   \[
   ES = 1283.8 - 323.4 = 960.4
   \]

Thus, Ben’s economic surplus from consuming tea is **HKD 960.40**.

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Do you have any questions or need further details?

Here are 5 related questions to expand on:
1. What would happen to Ben’s surplus if the price of tea increased to HKD 50 per cup?
2. How would the economic surplus change if the total benefit function changed?
3. What is the interpretation of marginal benefit in this scenario?
4. How do you calculate consumer surplus in cases with multiple goods?
5. How does diminishing marginal benefit affect consumption decisions?

**Tip:** Marginal benefit helps determine the optimal quantity of goods consumed, as long as it exceeds the price.

The equation below shows Ben’s benefit of consuming n cups of tea, and tea is perfectly divisible and can be consumed at any arbitrary quantity (say, 3.14 cups). Total benefit: TB(n) = 229n − 10n^2, Average benefit: AB(n) = 229 − 10n, Marginal benefit: MB(n) = 229 − 20n. Suppose the price is HKD 33 per cup. Ben’s economic surplus from consuming tea would be HKD [**Answer**].

Learn how to calculate Ben's economic surplus from consuming tea using the marginal benefit and total cost approach. This problem involves key algebraic concepts and marginal analysis suitable for students in Grades 10-12.

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