Answer :
Certainly! To graphically represent the system of inequalities:
[tex]\[ \begin{cases} y \geq 4x \\ y \leq 8.75 - 0.5\pi x^2 \end{cases} \][/tex]
we can follow these steps for a comprehensive solution.
### Step 1: Understand and Simplify the Inequalities
1. First Inequality: [tex]\( y \geq 4x \)[/tex].
This is a linear inequality representing a line with slope 4 and passing through the origin. The inequality indicates that the area above (or on) this line should be shaded.
2. Second Inequality: [tex]\( y \leq 8.75 - 0.5\pi x^2 \)[/tex].
This is a quadratic inequality representing an inverted parabola with its vertex at [tex]\( (0, 8.75) \)[/tex]. The inequality indicates that the area below (or on) this parabola should be shaded.
### Step 2: Plot the Boundary Lines and Parabola
- Linear Boundary (Line): [tex]\( y = 4x \)[/tex].
- This line can be determined by plotting points. For example:
- When [tex]\( x = -2 \)[/tex], [tex]\( y = -8 \)[/tex].
- When [tex]\( x = 0 \)[/tex], [tex]\( y = 0 \)[/tex].
- When [tex]\( x = 2 \)[/tex], [tex]\( y = 8 \)[/tex].
- Therefore, our line passes through these points: [tex]\((-10, -40)\)[/tex] to [tex]\( (10, 40) \)[/tex].
- Quadratic Boundary (Parabola): [tex]\( y = 8.75 - 0.5\pi x^2 \)[/tex].
- We should evaluate key points across a range of [tex]\( x \)[/tex] values:
- When [tex]\( x = -2 \)[/tex], [tex]\( y \approx -148.33 \)[/tex].
- When [tex]\( x = 0 \)[/tex], [tex]\( y = 8.75 \)[/tex].
- When [tex]\( x = 2 \)[/tex], [tex]\( y \approx -148.33 \)[/tex].
### Step 3: Define the Region of Intersection
- Identify the region that satisfies both inequalities. This will be the region:
- Above the line [tex]\( y = 4x \)[/tex] (since [tex]\( y \geq 4x \)[/tex])
- Below the curve [tex]\( y = 8.75 - 0.5\pi x^2 \)[/tex] (since [tex]\( y \leq 8.75 - 0.5\pi x^2 \)[/tex])
### Step 4: Sketch the Graph
1. Draw the line [tex]\( y = 4x \)[/tex]:
- Start from [tex]\( (-10, -40) \)[/tex] and go through the points up to [tex]\( (10, 40) \)[/tex].
2. Draw the parabola [tex]\( y = 8.75 - 0.5\pi x^2 \)[/tex]:
- Start from values of [tex]\( x \)[/tex] ranging approximately from [tex]\(-10\)[/tex] to [tex]\(10\)[/tex] and draft the curve passing through key points like [tex]\((0, 8.75)\)[/tex] and [tex]\((\pm2, \approx -148.33)\)[/tex].
3. Shade the region that lies above the line [tex]\( y = 4x \)[/tex] and below the parabola [tex]\( y = 8.75 - 0.5\pi x^2 \)[/tex].
### Step 5: Verify the Shaded Region
To confirm the region, check:
- Points that lie in the overlap should satisfy both inequalities.
For example:
- Test [tex]\( x = -0.5 \)[/tex]:
- [tex]\( y = 4(-0.5) = -2 \)[/tex]
- [tex]\( y = 8.75 - 0.5\pi(-0.5)^2 \approx 8.75 - 0.5\cdot0.5\pi \approx 8.75 - 0.785 \approx 7.965 \)[/tex]
- Check [tex]\( y \)[/tex] values between [tex]\(-2\)[/tex] and [tex]\( 7.965 \)[/tex].
### Step 6: Interpret the Results
- The shaded region provides all the [tex]\( (x, y) \)[/tex] pairs that satisfy the system of inequalities:
- The solutions start from a region defined roughly around the curve and line intersection.
### Conclusion
Given the inequalities and boundary curves provided:
[tex]\[ y = 4x \quad \text{and} \quad y = 8.75 - 0.5\pi x^2 \][/tex]
we have determined the graph, highlighting the solution region that lies above [tex]\( y = 4x \)[/tex] and below [tex]\( y = 8.75 - 0.5\pi x^2 \)[/tex]. For detailed exact numerical values, please refer directly to the computation results provided or the graph sketch for precise boundaries.
[tex]\[ \begin{cases} y \geq 4x \\ y \leq 8.75 - 0.5\pi x^2 \end{cases} \][/tex]
we can follow these steps for a comprehensive solution.
### Step 1: Understand and Simplify the Inequalities
1. First Inequality: [tex]\( y \geq 4x \)[/tex].
This is a linear inequality representing a line with slope 4 and passing through the origin. The inequality indicates that the area above (or on) this line should be shaded.
2. Second Inequality: [tex]\( y \leq 8.75 - 0.5\pi x^2 \)[/tex].
This is a quadratic inequality representing an inverted parabola with its vertex at [tex]\( (0, 8.75) \)[/tex]. The inequality indicates that the area below (or on) this parabola should be shaded.
### Step 2: Plot the Boundary Lines and Parabola
- Linear Boundary (Line): [tex]\( y = 4x \)[/tex].
- This line can be determined by plotting points. For example:
- When [tex]\( x = -2 \)[/tex], [tex]\( y = -8 \)[/tex].
- When [tex]\( x = 0 \)[/tex], [tex]\( y = 0 \)[/tex].
- When [tex]\( x = 2 \)[/tex], [tex]\( y = 8 \)[/tex].
- Therefore, our line passes through these points: [tex]\((-10, -40)\)[/tex] to [tex]\( (10, 40) \)[/tex].
- Quadratic Boundary (Parabola): [tex]\( y = 8.75 - 0.5\pi x^2 \)[/tex].
- We should evaluate key points across a range of [tex]\( x \)[/tex] values:
- When [tex]\( x = -2 \)[/tex], [tex]\( y \approx -148.33 \)[/tex].
- When [tex]\( x = 0 \)[/tex], [tex]\( y = 8.75 \)[/tex].
- When [tex]\( x = 2 \)[/tex], [tex]\( y \approx -148.33 \)[/tex].
### Step 3: Define the Region of Intersection
- Identify the region that satisfies both inequalities. This will be the region:
- Above the line [tex]\( y = 4x \)[/tex] (since [tex]\( y \geq 4x \)[/tex])
- Below the curve [tex]\( y = 8.75 - 0.5\pi x^2 \)[/tex] (since [tex]\( y \leq 8.75 - 0.5\pi x^2 \)[/tex])
### Step 4: Sketch the Graph
1. Draw the line [tex]\( y = 4x \)[/tex]:
- Start from [tex]\( (-10, -40) \)[/tex] and go through the points up to [tex]\( (10, 40) \)[/tex].
2. Draw the parabola [tex]\( y = 8.75 - 0.5\pi x^2 \)[/tex]:
- Start from values of [tex]\( x \)[/tex] ranging approximately from [tex]\(-10\)[/tex] to [tex]\(10\)[/tex] and draft the curve passing through key points like [tex]\((0, 8.75)\)[/tex] and [tex]\((\pm2, \approx -148.33)\)[/tex].
3. Shade the region that lies above the line [tex]\( y = 4x \)[/tex] and below the parabola [tex]\( y = 8.75 - 0.5\pi x^2 \)[/tex].
### Step 5: Verify the Shaded Region
To confirm the region, check:
- Points that lie in the overlap should satisfy both inequalities.
For example:
- Test [tex]\( x = -0.5 \)[/tex]:
- [tex]\( y = 4(-0.5) = -2 \)[/tex]
- [tex]\( y = 8.75 - 0.5\pi(-0.5)^2 \approx 8.75 - 0.5\cdot0.5\pi \approx 8.75 - 0.785 \approx 7.965 \)[/tex]
- Check [tex]\( y \)[/tex] values between [tex]\(-2\)[/tex] and [tex]\( 7.965 \)[/tex].
### Step 6: Interpret the Results
- The shaded region provides all the [tex]\( (x, y) \)[/tex] pairs that satisfy the system of inequalities:
- The solutions start from a region defined roughly around the curve and line intersection.
### Conclusion
Given the inequalities and boundary curves provided:
[tex]\[ y = 4x \quad \text{and} \quad y = 8.75 - 0.5\pi x^2 \][/tex]
we have determined the graph, highlighting the solution region that lies above [tex]\( y = 4x \)[/tex] and below [tex]\( y = 8.75 - 0.5\pi x^2 \)[/tex]. For detailed exact numerical values, please refer directly to the computation results provided or the graph sketch for precise boundaries.
