Problem 28 - Precalculus: Mathematical for Calculus

Question

Graph the solution of the system of inequalities. Find the coordinates of all vertices, and determine whether the solution set is bounded. $$\left\\{\begin{array}{r}x>0 \\\y>0 \\\x+y<10 \\\x^{2}+y^{2}>9\end{array}\right.$$

Step-by-Step Solution

Answer

Vertices: (3, 7) and (7, 3); Solution set is bounded.

Step 1: Identify Boundaries of Inequalities

The system of inequalities involves four different conditions. The first two, \(x > 0\) and \(y > 0\), restrict the solution to the first quadrant. The third inequality, \(x + y < 10\), represents all points below the line \(x + y = 10\). The fourth inequality, \(x^2 + y^2 > 9\), represents points outside the circle with radius 3 centered at the origin \((0, 0)\). Our task is to find the region that satisfies all these conditions.

Step 2: Sketch Key Boundaries

Begin by sketching the boundary lines on a graph. The line \(x + y = 10\) is a straight line with intercepts at \((10, 0)\) and \((0, 10)\). The circle \(x^2 + y^2 = 9\) is a circle centered at \((0, 0)\) with a radius of 3. These boundaries, along with the axes from \(x > 0\) and \(y > 0\), form the framework of your plot.

Step 3: Identify Solution Region

The solution region must satisfy all inequalities. Shade the region below the line \(x + y = 10\) that falls outside the circle \(x^2 + y^2 = 9\) and still resides in the first quadrant. This effectively creates a crescent-like area in the top right part of the coordinate plane that isn't directly adjacent to the axes.

Step 4: Find the Vertices

Identify the corner points of this shaded region. Since the jagged edge is the intersection between the line \(x + y = 10\) and the circle \(x^2 + y^2 = 9\), solve these equations simultaneously. Substitute \(x = 10 - y\) into \(x^2 + y^2 = 9\), yielding \((10-y)^2 + y^2 = 9\). Solve for \(y\) to get the intersection points. These calculations result in the two vertices where \(x + y = 10\) meets the circle: approximately \((3, 7)\) and \((7, 3)\).

Step 5: Determine Boundedness of the Solution

The solution set is bounded if it is enclosed within a finite region. Although the shaded region covers an area under the line \(x + y = 10\), it's limited by the circle \(x^2 + y^2 = 9\). This crescent-shaped area between them does have clear boundaries, hence the solution set is bounded.

Key Concepts

Graphing Inequalities

When graphing inequalities, the first step is to determine the boundaries created by each inequality. In our example, we have four inequalities to consider.

• The inequalities \(x > 0\) and \(y > 0\) tell us that we are only considering the first quadrant of the coordinate plane, which is the top-right section where both x and y are positive.
• The inequality \(x + y < 10\) represents an area below the straight line \(x + y = 10\). This line crosses the axes at the points \((10,0)\) and \((0,10)\). Below this line is a triangular region stretching towards the origin.
• The inequality \(x^2 + y^2 > 9\) indicates an area outside a circle centered at the origin with a radius of 3, which can be thought of as the region beyond this circle's perimeter.

By graphing these lines and boundaries on a coordinate plane, we can identify where these regions overlap.

We shade this area to highlight the solution to the system of inequalities, which is the region that meets all conditions simultaneously.

Coordinate Plane

A coordinate plane is like a large, two-dimensional graph where you can plot points based on their x and y values. It consists of two axes: the horizontal axis (x-axis) and the vertical axis (y-axis).

• When plotting inequalities, the coordinate plane helps visually articulate the limits or constraints provided by each inequality.
• For instance, the inequality \(x > 0\) indicates that we're only looking at the regions to the right of the y-axis, and \(y > 0\) means the regions above the x-axis.

The first quadrant, where both \(x > 0\) and \(y > 0\), is critically important for this problem. It's where both of these inequalities are simultaneously satisfied.

Graphing on the coordinate plane allows us to see where the line \(x + y = 10\) and the boundary of the circle \(x^2 + y^2 = 9\) intersect this quadrant, which helps in identifying the solution set for the system of inequalities.

Bounded Solution

A solution is "bounded" when it fits within a finite area on the coordinate plane. In our case, the solution set is bounded because it falls within clear-cut borders.

• The boundary line \(x + y = 10\) and the circle \(x^2 + y^2 = 9\) create natural confines, with one being a straight line and the other, a curve originating from the same point.
• The first quadrant acts as a general border when combining these inequalities because \(x > 0\) and \(y > 0\) restrict the region to positive x and y values.

This creates a shape that looks like a crescent moon where the constraints meet. Because this region doesn't extend infinitely in any direction due to its borders, the solution is considered bounded.

In simpler terms, if you can "capture" the solution region using a finite, imaginary loop drawn on the coordinate plane, it is bounded. In the context of our exercise, this is exactly what happens.