Which one of the following set of constraints does the given shaded region represent?

The shaded region is bounded by several lines. Each line forms a boundary, and each inequality determines which side of that line is included in the region.

The boundary lines are:

• Two parallel diagonal lines: $x + y = 15$ (lower) and $x + y = 30$ (upper)
• A vertical line: $x = 15$ (on the right side)
• A horizontal line: $y = 20$ (on the top)
• The axes: $x = 0$ and $y = 0$ (first quadrant)

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For $x + y = 15$:

The shaded region is above this line.

Constraint: $x + y \geq 15$

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For $x + y = 30$:

The shaded region is below this line.

Constraint: $x + y \leq 30$

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For $x = 15$:

The shaded region is to the left of this line.

Constraint: $x \leq 15$

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For $y = 20$:

The shaded region is below this line.

Constraint: $y \leq 20$

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For the axes:

The shaded region is in the first quadrant.

Constraint: $x, y \geq 0$

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The complete set of constraints:

• $x + y \leq 30$
• $x + y \geq 15$
• $x \leq 15$
• $y \leq 20$
• $x, y \geq 0$

This matches Option 1.

Options 3 and 4 contain $x + y \geq 30$ and $x + y \leq 15$, which cannot be satisfied simultaneously.

Therefore, the answer is Option 1: $x + y \leq 30, x + y \geq 15, x \leq 15, y \leq 20, x, y \geq 0$