Answer :
To determine the region corresponding to the values of [tex]\(x\)[/tex] (the area in square feet used for flowers) and [tex]\(y\)[/tex] (the area in square feet used for vegetables) that satisfy the gardener's requirements, we need to consider the given conditions:
1. The total area used for planting flowers and vegetables can be up to 230 square feet.
2. The area used for vegetables must be at least four times the area used for flowers.
We can translate these conditions into mathematical inequalities:
1. [tex]\(x + y \leq 230\)[/tex] (the total area constraint)
2. [tex]\(y \geq 4x\)[/tex] (the constraint that the area used for vegetables is at least four times the area used for flowers)
Next, we will identify the boundary lines for these inequalities and their intersections with the axes.
### Step-by-Step Solution:
1. Identify the Boundary for [tex]\(x + y \leq 230\)[/tex]:
- When [tex]\(x = 0\)[/tex], solving for [tex]\(y\)[/tex] gives:
[tex]\[ y = 230 \][/tex]
This point is [tex]\((0, 230)\)[/tex].
- When [tex]\(y = 0\)[/tex], solving for [tex]\(x\)[/tex] gives:
[tex]\[ x = 230 \][/tex]
This point is [tex]\((230, 0)\)[/tex].
Therefore, the line [tex]\(x + y = 230\)[/tex] intersects the [tex]\(x\)[/tex]-axis at [tex]\((230, 0)\)[/tex] and the [tex]\(y\)[/tex]-axis at [tex]\((0, 230)\)[/tex].
2. Identify the Boundary for [tex]\(y \geq 4x\)[/tex]:
- When [tex]\(x = 0\)[/tex], solving for [tex]\(y\)[/tex] gives:
[tex]\[ y = 0 \][/tex]
This point is [tex]\((0, 0)\)[/tex].
- To find another point, consider where this line intersects the line [tex]\(x + y = 230\)[/tex]. Set [tex]\(y = 4x\)[/tex] in the equation [tex]\(x + y = 230\)[/tex]:
[tex]\[ x + 4x = 230 \implies 5x = 230 \implies x = 46 \][/tex]
Then,
[tex]\[ y = 4 \cdot 46 = 184 \][/tex]
This point is [tex]\((46, 184)\)[/tex].
So, the line [tex]\(y = 4x\)[/tex] intersects the origin at [tex]\((0, 0)\)[/tex] and at [tex]\((46, 184)\)[/tex] on the line [tex]\(x + y = 230\)[/tex].
### Visual Representation:
Using these boundary points, the feasible region is defined by the area bounded by the lines:
- The line [tex]\(x + y = 230\)[/tex] with endpoints [tex]\((0, 230)\)[/tex] and [tex]\((230, 0)\)[/tex].
- The line [tex]\(y = 4x\)[/tex] with endpoints [tex]\((0, 0)\)[/tex] and [tex]\((46, 184)\)[/tex],
We shade the region that satisfies both inequalities:
- Below the line [tex]\(x + y \leq 230\)[/tex].
- Above the line [tex]\(y \geq 4x\)[/tex].
This shaded region on the graph would show the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] that satisfy the gardener’s requirements for planting flowers and vegetables.
1. The total area used for planting flowers and vegetables can be up to 230 square feet.
2. The area used for vegetables must be at least four times the area used for flowers.
We can translate these conditions into mathematical inequalities:
1. [tex]\(x + y \leq 230\)[/tex] (the total area constraint)
2. [tex]\(y \geq 4x\)[/tex] (the constraint that the area used for vegetables is at least four times the area used for flowers)
Next, we will identify the boundary lines for these inequalities and their intersections with the axes.
### Step-by-Step Solution:
1. Identify the Boundary for [tex]\(x + y \leq 230\)[/tex]:
- When [tex]\(x = 0\)[/tex], solving for [tex]\(y\)[/tex] gives:
[tex]\[ y = 230 \][/tex]
This point is [tex]\((0, 230)\)[/tex].
- When [tex]\(y = 0\)[/tex], solving for [tex]\(x\)[/tex] gives:
[tex]\[ x = 230 \][/tex]
This point is [tex]\((230, 0)\)[/tex].
Therefore, the line [tex]\(x + y = 230\)[/tex] intersects the [tex]\(x\)[/tex]-axis at [tex]\((230, 0)\)[/tex] and the [tex]\(y\)[/tex]-axis at [tex]\((0, 230)\)[/tex].
2. Identify the Boundary for [tex]\(y \geq 4x\)[/tex]:
- When [tex]\(x = 0\)[/tex], solving for [tex]\(y\)[/tex] gives:
[tex]\[ y = 0 \][/tex]
This point is [tex]\((0, 0)\)[/tex].
- To find another point, consider where this line intersects the line [tex]\(x + y = 230\)[/tex]. Set [tex]\(y = 4x\)[/tex] in the equation [tex]\(x + y = 230\)[/tex]:
[tex]\[ x + 4x = 230 \implies 5x = 230 \implies x = 46 \][/tex]
Then,
[tex]\[ y = 4 \cdot 46 = 184 \][/tex]
This point is [tex]\((46, 184)\)[/tex].
So, the line [tex]\(y = 4x\)[/tex] intersects the origin at [tex]\((0, 0)\)[/tex] and at [tex]\((46, 184)\)[/tex] on the line [tex]\(x + y = 230\)[/tex].
### Visual Representation:
Using these boundary points, the feasible region is defined by the area bounded by the lines:
- The line [tex]\(x + y = 230\)[/tex] with endpoints [tex]\((0, 230)\)[/tex] and [tex]\((230, 0)\)[/tex].
- The line [tex]\(y = 4x\)[/tex] with endpoints [tex]\((0, 0)\)[/tex] and [tex]\((46, 184)\)[/tex],
We shade the region that satisfies both inequalities:
- Below the line [tex]\(x + y \leq 230\)[/tex].
- Above the line [tex]\(y \geq 4x\)[/tex].
This shaded region on the graph would show the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] that satisfy the gardener’s requirements for planting flowers and vegetables.
