Question
Mr. Jones needs to build a rectangular fence in his backyard to keep their dog safe. The length of the fence (y) should be at least 60 feet, and the distance around (perimeter) should be no more than 260 feet. Which system of inequalities and graph represent the possible dimensions of the pen?(1 point)
Responses
Responses
Answers
GPT-4o mini
To solve Mr. Jones's problem regarding the dimensions of the rectangular fence, we need to establish a system of inequalities based on the constraints provided.
Let:
- \( y \) = length of the fence
- \( x \) = width of the fence
From the information given:
1. The length \( y \) must be at least 60 feet:
\[
y \geq 60
\]
2. The perimeter of a rectangle is given by the formula \( P = 2x + 2y \). According to the problem, the perimeter should be no more than 260 feet:
\[
2x + 2y \leq 260
\]
This can be simplified by dividing the entire inequality by 2:
\[
x + y \leq 130
\]
Now we can summarize the system of inequalities:
\[
\begin{align*}
1. & \quad y \geq 60 \\
2. & \quad x + y \leq 130
\end{align*}
\]
Next, we can graph this system:
1. **Graphing \( y \geq 60 \)**:
- This is a horizontal line at \( y = 60 \). The area above this line (including the line itself) is the solution.
2. **Graphing \( x + y \leq 130 \)**:
- To graph the line \( x + y = 130 \), rearranging it gives \( y = -x + 130 \). This line has a y-intercept of 130 and an x-intercept of 130 (the points where it crosses the axes).
- The area below this line (including the line itself) is part of the solution.
3. **Finding the feasible region**:
- The solution set will be the area that satisfies both inequalities, which is where the shaded regions for both inequalities overlap.
In summary, the system of inequalities is:
\[
\begin{align*}
1. & \quad y \geq 60 \\
2. & \quad x + y \leq 130
\end{align*}
\]
And the graph representing the possible dimensions of the pen will include the area above the line \( y = 60 \) and below the line \( x + y = 130 \), creating a polygonal region where the feasible dimensions (width \( x \) and length \( y \)) can be determined.
Let:
- \( y \) = length of the fence
- \( x \) = width of the fence
From the information given:
1. The length \( y \) must be at least 60 feet:
\[
y \geq 60
\]
2. The perimeter of a rectangle is given by the formula \( P = 2x + 2y \). According to the problem, the perimeter should be no more than 260 feet:
\[
2x + 2y \leq 260
\]
This can be simplified by dividing the entire inequality by 2:
\[
x + y \leq 130
\]
Now we can summarize the system of inequalities:
\[
\begin{align*}
1. & \quad y \geq 60 \\
2. & \quad x + y \leq 130
\end{align*}
\]
Next, we can graph this system:
1. **Graphing \( y \geq 60 \)**:
- This is a horizontal line at \( y = 60 \). The area above this line (including the line itself) is the solution.
2. **Graphing \( x + y \leq 130 \)**:
- To graph the line \( x + y = 130 \), rearranging it gives \( y = -x + 130 \). This line has a y-intercept of 130 and an x-intercept of 130 (the points where it crosses the axes).
- The area below this line (including the line itself) is part of the solution.
3. **Finding the feasible region**:
- The solution set will be the area that satisfies both inequalities, which is where the shaded regions for both inequalities overlap.
In summary, the system of inequalities is:
\[
\begin{align*}
1. & \quad y \geq 60 \\
2. & \quad x + y \leq 130
\end{align*}
\]
And the graph representing the possible dimensions of the pen will include the area above the line \( y = 60 \) and below the line \( x + y = 130 \), creating a polygonal region where the feasible dimensions (width \( x \) and length \( y \)) can be determined.