Chapter 10.9, Problem 29E

To determine

To find: the coordinates of all the vertices and determine whether the solution set bounds or not.

Answer to Problem 29E

The vertices are $[(0,0),(3,0)\text{ and }(0,9)]$

The solution set is bounded

Explanation of Solution

Given:

  $\begin{array}{l}y\le 9-x^2\\ x\ge 0,y\ge 0,\end{array}$

Calculation:

We obtain the equation of the parabola from the given inequality as

  $y=9-x^2$

Some points on the parabola $y=9-x^2$ are

X	-4	-3	0	3	4
Y	-7	0	9	0	-7

Using these points we first draw the graph of this parabola.

We graph the parabola given by the equation that corresponds to the inequality. To determine the graph of the inequality, we need to check our test point. Here, we take (0,0) as test point.

Inequality	Test point (0,0)	Conclusion
$y\le 9-x^2$	$0\le 9$	Satisfies inequality

Since (0,0) is below the parabola $y=9-x^2,$ our check shows that the region on or below the parabola must satisfy the inequality. The inequalities $x\ge 0$ and $y\ge 0$ shows that $x$ and $y$ are nonnegative.

Therefore, graph with shaded region is

  

Vertices: - The origin $(0,0)$ is a vertex. The other two vertices are at the x and y intercept of the parabola in the first quadrant: $(3,0)\text{ and }(0,9)$ respectively.

Therefore, the vertices are $[(0,0),(3,0)\text{ and }(0,9)]$

From the figure it is clear that the shaded region is bounded. Therefore, the solution set is bounded

Conclusion:

Therefore, the vertices are $[(0,0),(3,0)\text{ and }(0,9)]$ and the solution set is bounded