(c) Based on the subdivisions in ​(a) and ​(b), describe a possible final fair division of the cake. Determine which pieces Carlos chooses. Select all that apply.

A.

60 degrees60°

vanilladash–20 degrees20°

strawberry

Your answer is not correct.

B.

180 degrees180°

​vanilla-only

C.

60 degrees60°

​strawberry-only

D.

20 degrees20°

vanilladash–60 degrees60°

strawberry

E.

120 degrees120°

​vanilla-only

This is the correct answer.

F.

28 degrees28°

​vanilla-only

G.

40 degrees40°

​strawberry-only

This is the correct answer.

H.

20 degrees20°

​strawberry-only

I.

90 degrees90°

​strawberry-only

J.

50 degrees50°

​strawberry-only

Part 5

​(d) For the final fair division found in part

​(c)​,

find the value​ (in dollars and​ cents) of each share in the eyes of the player receiving it.

What is the value of​ Angela's share?

To tackle this problem, let's first break down how to solve part **(d)** by focusing on the final fair division based on part **(c)**.

In part **(c)**, the correct divisions of the cake that Carlos chooses are:

- **E. 120 degrees** (vanilla-only)
- **G. 40 degrees** (strawberry-only)

These choices imply Carlos values a total of **120 degrees** of vanilla and **40 degrees** of strawberry.

Now, for part **(d)**, you are asked to find the value of Angela's share. Here's the approach:

### 1. Total cake value and share
Assume the entire cake is worth a certain amount (for example, $100) for simplicity. Since the cake is a full 360 degrees, each degree of the cake is worth $ \frac{100}{360} $ dollars, or approximately 0.2778 dollars per degree.

### 2. Angela's share
To calculate the value of Angela's share, we need to identify the specific slices left for her, based on the pieces Carlos has already chosen. 

Carlos took:

- **120 degrees** of vanilla
- **40 degrees** of strawberry

So Angela receives whatever is left from the cake. The remaining cake for Angela would consist of:

- **60 degrees** of vanilla (since 180 degrees of vanilla existed in total, and Carlos took 120 degrees of it)
- **60 degrees** of strawberry (since 100 degrees of strawberry existed in total, and Carlos took 40 degrees of it)

### 3. Value of Angela's share
To find the value of Angela’s share, we can sum up the dollar value for the pieces of cake she receives:

- **60 degrees of vanilla**: $ 60 \times 0.2778 = 16.67 $ dollars
- **60 degrees of strawberry**: $ 60 \times 0.2778 = 16.67 $ dollars

Therefore, the total value of Angela's share is:

\[
16.67 + 16.67 = 33.34 \text{ dollars}
\]

### Final Answer:
The value of Angela's share is **$33.34**.

---

Would you like more details or have any questions?

### Follow-up Questions:
1. How would the total value change if the cake were worth $200 instead of $100?
2. What if Carlos preferred strawberry more than vanilla—how would his choices change?
3. Could Angela and Carlos trade pieces to reach a better division for both of them?
4. How does changing the size of each slice impact the fairness of the division?
5. What would the value of each slice be if the cake were divided into more than 360 degrees?

### Tip:
When dividing resources like cake, assigning values to different parts can help quantify fairness and find equitable solutions.

Based on the subdivisions in ​(a) and ​(b), describe a possible final fair division of the cake. Determine which pieces Carlos chooses. Select all that apply.

A. 60 degrees (vanilla) – 20 degrees (strawberry)
B. 180 degrees (vanilla-only)
C. 60 degrees (strawberry-only)
D. 20 degrees (vanilla) – 60 degrees (strawberry)
E. 120 degrees (vanilla-only)
F. 28 degrees (vanilla-only)
G. 40 degrees (strawberry-only)
H. 20 degrees (strawberry-only)
I. 90 degrees (strawberry-only)
J. 50 degrees (strawberry-only)

For the final fair division found in part (c), find the value (in dollars and cents) of each share in the eyes of the player receiving it. What is the value of Angela's share?

This problem demonstrates how to calculate the value of Angela's share using fair division principles. Given specific cake divisions based on angles, students will apply proportional allocation methods to find each player's share. Suitable for Grades 10-12.

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