David and Paula are planning to divide the​ chocolate-vanilla-strawberry cake shown on the right using the​ divider-chooser method. David likes chocolate cake and vanilla cake the​ same, and he likes both of them four timesfour times as much as strawberry cake. Paula likes vanilla cake and strawberry cake the​ same, but she is allergic to chocolate cake. Suppose that David is the divider and Paula is the chooser. Complete parts​ (a) through​ (c) below.
 


A circular cake is divided into 3 equal sections by three line segments, each of which has the circle's center as an endpoint.
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Part 1
​(a) Is the cut shown in the figure on the right a possible​ 50-50 cut that David might have made as the​ divider? If​ so, describe the share Paula should choose and give the value​ (as a​ percent) of that share to Paula. If the cut is not a​ 50-50 cut, give the values of the two shares to David. Select the correct choice below and fill in the answer​ box(es) within your choice.
​(Round to the nearest percent as​ needed.)
A.
​No; to​ David, s 1s1 is worth enter your response here​% and s 2s2 is worth enter your response here​%.
B.
​Yes; Paula chooses s 2s2 and its value is enter your response here​%.
C.
​Yes; Paula chooses s 1s1 and its value is enter your response here​%.

To solve this problem, let's first analyze David's and Paula's preferences and calculate the value of the cake pieces for both of them.

### Step 1: Understanding Preferences
- **David's Preferences:**
  - David likes chocolate and vanilla cake equally.
  - He values both chocolate and vanilla **four times** as much as strawberry cake.
  - If we let the value of the strawberry cake be \(1\), then the value of chocolate or vanilla is \(4\).
  
- **Paula's Preferences:**
  - Paula likes vanilla and strawberry cake equally.
  - She is **allergic** to chocolate cake, so she assigns it a value of \(0\).
  - If we let the value of the strawberry cake be \(1\), then the value of vanilla cake is also \(1\).

### Step 2: Assigning Values to Cake Sections
Assume the cake is divided into three equal parts:
- One section is chocolate.
- One section is vanilla.
- One section is strawberry.

Since each section is \( \frac{1}{3} \) of the cake:
- **David's Valuation:**
  - Value of chocolate section = \(4 \times \frac{1}{3} = \frac{4}{3}\)
  - Value of vanilla section = \(4 \times \frac{1}{3} = \frac{4}{3}\)
  - Value of strawberry section = \(1 \times \frac{1}{3} = \frac{1}{3}\)

  Total value for David = \( \frac{4}{3} + \frac{4}{3} + \frac{1}{3} = \frac{9}{3} = 3 \) units.

- **Paula's Valuation:**
  - Value of chocolate section = \(0 \times \frac{1}{3} = 0\)
  - Value of vanilla section = \(1 \times \frac{1}{3} = \frac{1}{3}\)
  - Value of strawberry section = \(1 \times \frac{1}{3} = \frac{1}{3}\)

  Total value for Paula = \( 0 + \frac{1}{3} + \frac{1}{3} = \frac{2}{3} \) units.

### Step 3: Evaluate the Cut
Suppose David makes the cut such that:
- \(s_1\) contains the chocolate and vanilla sections.
- \(s_2\) contains the strawberry section.

- **David's Valuation:**
  - \(s_1\) (chocolate and vanilla) = \( \frac{4}{3} + \frac{4}{3} = \frac{8}{3} \) units.
  - \(s_2\) (strawberry) = \( \frac{1}{3} \) unit.

- **Paula's Valuation:**
  - \(s_1\) (chocolate and vanilla) = \( 0 + \frac{1}{3} = \frac{1}{3} \) unit.
  - \(s_2\) (strawberry) = \( \frac{1}{3} \) unit.

### Step 4: Conclusion
- \(s_1\) is worth \( \frac{8}{3} \) for David, and \(s_2\) is worth \( \frac{1}{3} \).
- Paula values \(s_1\) at \( \frac{1}{3} \) and \(s_2\) also at \( \frac{1}{3} \).

Given the valuations, the cut is **not** a 50-50 cut according to David's perspective. Here's why:
- \(s_1\) is worth \( \frac{8}{3} \) for David, which is approximately \( 89\%\) of his total valuation.
- \(s_2\) is worth \( \frac{1}{3} \), which is approximately \(11\%\) of his total valuation.

### Final Answer
So, the correct choice is **A**:
- \(s_1\) is worth approximately 89% to David.
- \(s_2\) is worth approximately 11% to David.

Let me know if you want further details or have any questions! 

Here are 5 related questions:
1. How would Paula's share be valued if she were the divider?
2. If Paula's allergic reaction to chocolate were less severe, how would it affect her choice?
3. What if David valued vanilla cake twice as much as chocolate cake?
4. How would the division change if David and Paula were to negotiate a different method than the divider-chooser?
5. What is a 50-50 cut in this context, and how can David ensure it?

**Tip:** When applying the divider-chooser method, the divider should aim to split the items so that the chooser values both parts equally, ensuring fairness.

David and Paula are planning to divide the chocolate-vanilla-strawberry cake shown on the right using the divider-chooser method. David likes chocolate cake and vanilla cake the same, and he likes both of them four times as much as strawberry cake. Paula likes vanilla cake and strawberry cake the same, but she is allergic to chocolate cake. Suppose that David is the divider and Paula is the chooser. Complete parts (a) through (c) below.

Learn how to apply the divider-chooser method for fair division of a chocolate-vanilla-strawberry cake between two individuals with different preferences. Understand key concepts of game theory, calculate values, and find the best strategies for division. Suitable for high school students in Grades 9-12.

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