Math Problem Statement

Explanation:

Step 1: Define the inequalities from the problem.

The problem states that the gardener will use up to 250 square feet for both flowers, $x$, and vegetables, $y$: $x+y \le 250$. Additionally, the area used for vegetables, $y$, should be at least twice the area used for flowers, $x$: $y \ge 2x$. Lastly, since we are considering area, we know that both $x$ and $y$ must be greater than or equal to zero: $x\ge0$ and $y\ge0$.

Step 2: Convert the inequalities into equations to plot the boundary lines.

Replace the inequality symbols with equal signs to find the equations of the lines: $x+y=250$, $y=2x$, $x=0$, $y=0$.

Step 3: Plot the boundary lines on the coordinate plane.

  • For $x + y = 250$, when $x=0$, then $y=250$, and when $y=0$, then $x=250$. Connect points $(0,250)$ and $(250,0)$ to graph the line, noting that the intercepts are outside of the provided graph.
  • For $y=2x$, when $x=0$, then $y=0$, and when $x=100$, then $y=200$. Connect points $(0,0)$ and $(100,200)$ to graph the line.
  • $x=0$ is the same as the y-axis.
  • $y=0$ is the same as the x-axis.

Step 4: Determine the shading for each inequality.

  • For $x+y\le250$, test point $(0,0)$ in the inequality. Since $0+0\le250$ is true, shade the region below the line $x+y=250$ that contains the point $(0,0)$.
  • For $y\ge2x$, test point $(100,0)$. Since $0\ge2(100)$ is false, shade the region above the line $y=2x$ that does not contain $(100,0)$.
  • For $x\ge0$ and $y\ge0$, shade the region in the first quadrant where both $x$ and $y$ are positive.

Step 5: Identify the overlapping shaded region.

The solution to the system of inequalities is the region where all shaded areas overlap. This will be a triangular region bounded by the lines $y=2x$, $x+y=250$, and the y-axis.

Answer:

The shaded region on the graph below represents all possible combinations of area for flowers ($x$) and vegetables ($y$) that satisfy the gardener's requirements.

Graph of the solution

Solution

Your explanation thoroughly outlines the steps to solve the problem involving the gardener's use of space for flowers and vegetables. Let's summarize the key points and review the solution:

  1. Inequalities from the Problem:

    • The gardener can use up to 250 square feet: x+y250x + y \leq 250.
    • The area for vegetables should be at least twice that for flowers: y2xy \geq 2x.
    • Both areas must be non-negative: x0x \geq 0 and y0y \geq 0.
  2. Equations for Boundary Lines:

    • Convert the inequalities to equations:
      • x+y=250x + y = 250
      • y=2xy = 2x
      • x=0x = 0
      • y=0y = 0
  3. Plotting the Boundary Lines:

    • Plot the line x+y=250x + y = 250 by connecting points (0,250)(0, 250) and (250,0)(250, 0).
    • Plot the line y=2xy = 2x by connecting points (0,0)(0, 0) and (100,200)(100, 200).
    • The lines x=0x = 0 and y=0y = 0 correspond to the y-axis and x-axis, respectively.
  4. Shading for the Inequalities:

    • For x+y250x + y \leq 250, the region below the line is shaded.
    • For y2xy \geq 2x, the region above the line is shaded.
    • The region in the first quadrant where x0x \geq 0 and y0y \geq 0 is shaded.
  5. Solution Region:

    • The triangular region where the shaded areas overlap is the solution, representing all possible combinations of areas for flowers and vegetables that meet the gardener's criteria.

This triangular region is bound by the lines y=2xy = 2x, x+y=250x + y = 250, and the y-axis (where x=0x = 0).

Would you like more details or have any questions about the steps?

Here are some additional questions you might find interesting:

  1. What happens if the gardener decides to use more than 250 square feet?
  2. How would the solution change if the condition y2xy \geq 2x were replaced by y2xy \leq 2x?
  3. What is the maximum area that can be used for flowers while still satisfying all conditions?
  4. How would the inequalities change if the gardener wanted the area for vegetables to be exactly twice that for flowers?
  5. Can you determine the exact coordinates of the vertices of the triangular solution region?

Tip: Always verify your shaded regions by testing points on both sides of your boundary lines. This ensures that the correct area is identified for each inequality.