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The problem you're working on involves a set \( S \), formed by the intersection of various geometric regions \( T \), \( U \), and \( P \), and contains three parts that build upon one another. Here's the breakdown:

### Problem (5 points) Consider the set:

\[
S = \{(x, y) \in \mathbb{R}^2 : (x - 14)^2 + y^2 \leq 110, \, (y - 14)^2 + x^2 \leq 100, \, x + y \leq 14 \} = T \cap U \cap P
\]

Where:
- \( T = \{(x, y) \in \mathbb{R}^2 : (x - 14)^2 + y^2 \leq 110 \} \) (a disk),
- \( U = \{(x, y) \in \mathbb{R}^2 : (y - 14)^2 + x^2 \leq 100 \} \) (another disk),
- \( P = \{(x, y) \in \mathbb{R}^2 : x + y \leq 14 \} \) (a half-plane).

### Part (a): **Describe the set \( S \) and the intersections \( T \cap U \), \( T \cap P \), and \( U \cap P \)**

- **Set \( S \)**: It's the intersection of two disks and a half-plane, geometrically representing a region bound by the overlapping areas of these shapes.
- **Intersection of \( T \cap U \)**: The disks \( T \) and \( U \) will overlap, as both are centered near each other and have similar radii.
- **Intersection of \( T \cap P \)** and **\( U \cap P \)**: These will be portions of the disks that lie beneath the line \( x + y = 14 \).

### Part (b): **Determine the boundary points where \( T \), \( U \), and \( P \) intersect, and draw the set \( S \) neatly.**

- **Finding Boundary Points**: To determine the points where the boundaries of \( T \), \( U \), and \( P \) intersect, we would need to solve the system of equations given by the boundaries of \( T \), \( U \), and \( P \). These boundaries are derived from the equations of the circles and the line \( x + y = 14 \).
  
- **Drawing the Set \( S \)**: After finding these intersections, you could sketch two overlapping disks with the line cutting off part of the intersection area. The points where the boundaries of all three shapes meet will give critical points to define the boundary of \( S \).

### Part (c): **Find the points that minimize the function \( f(x, y) = x + y \) over the set \( S \).**

- To minimize \( f(x, y) = x + y \), you will likely need to evaluate this function along the boundaries of \( S \), especially focusing on the points of intersection found in part (b), as these points often give local minima in constrained optimization problems.

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Would you like me to work out the solution for the system of equations in part (b) or provide more details on how to minimize \( f(x, y) \)? Let me know!

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Here are 5 related questions you might find helpful:
1. How do you determine whether two disks intersect?
2. What geometric shapes can result from the intersection of a disk and a half-plane?
3. How can optimization be applied when the objective function is linear?
4. What methods are used to solve systems of equations involving circle and line intersections?
5. How do you find the boundary of a set formed by the intersection of regions?

**Tip**: When solving systems of equations involving circles, try using substitution or elimination to simplify the process.

Consider the set S = {(x, y) in R^2: (x - 14)^2 + y^2 <= 110, (y - 14)^2 + x^2 <= 100, x + y <= 14}, and describe its intersections and minimization.

Explore the problem of finding the minimum of f(x, y) = x + y over a set defined by the intersection of two disks and a half-plane. Understand the geometry and optimization methods involved.

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