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Chapter 6: Problem 64

Concept Check Fill in each blank with the appropriate response. The graph of the system $$ y > x^{2}+2 $$ $$ \begin{array}{r} x^{2}+y^{2}<16 \\ y<7 \end{array} $$ consists of all points \(\frac{\text { the parabola given by }}{\text { (above/below) }}\) \(y=x^{2}+2, \frac{ }{\text { (inside/outside) }}\) the circle \(x^{2}+y^{2}=16,\) and (above/below) the line \(y =7\)

Short Answer

Expert verified
Above the parabola, inside the circle, below the line.

Step by step solution

01

Identify the Parabola and its Region

The inequality \(y > x^2 + 2\) defines the region above the parabola \(y = x^2 + 2\). This means any point that satisfies the inequality is located above this parabola in the coordinate plane.
02

Identify the Circle and its Region

The inequality \(x^2 + y^2 < 16\) represents the region inside the circle centered at the origin with a radius of 4 (since \(16 = 4^2\)). Points satisfying this inequality are within this circle.
03

Identify the Line and its Region

The inequality \(y < 7\) indicates the region below the horizontal line \(y = 7\). This means that points satisfying this inequality lie below this line.
04

Combine the Conditions

The system of inequalities describes the solution set for points that are simultaneously: 1. Above the parabola \(y = x^2 + 2\), 2. Inside the circle \(x^2 + y^2 = 16\), and 3. Below the line \(y = 7\). Thus, the combined solution set is defined by these spatial relations to the given figures.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Parabola Region
When dealing with a parabola like the one given by the equation \( y = x^2 + 2 \), it's important to understand what "above the parabola" means in the context of an inequality. The inequality \( y > x^2 + 2 \) tells us that we're considering all the points in the coordinate plane that lie above the curve of this parabola. Imagine drawing the parabola on the graph as a solid curve, and colors all the area above it. This area includes any point where the y-coordinate is greater than that on the curve.

Here are some features to remember:
* The vertex of this parabola is at the point \( (0, 2) \).
* Since it opens upwards (because of the positive sign in front of \( x^2 \)), any y-value greater than the parabola itself satisfies the inequality.

Understanding this lets us determine which parts of the graph can potentially satisfy the other inequalities as well.
Circle Inequality
The circle is defined by the equation \( x^2 + y^2 = 16 \). In the context of an inequality like \( x^2 + y^2 < 16 \), we are interested in the region inside this circle. Think of the circle as having a boundary, and the inequality captures all the points strictly inside this boundary, not on it. This means every point that falls inside the circle satisfies this inequality.

Here鈥檚 how to understand it better:
* The circle is centered at the origin point \( (0,0) \).
* It has a radius of 4, because \( 4^2 = 16 \).
* Every point whose distance from the origin is less than 4 units is part of this region.

Visualizing each inequality separately helps in understanding how they might overlap or combine with other regions described by different inequalities.
Line Inequality
A line like \( y = 7 \) creates a horizontal boundary on the graph. When presented with \( y < 7 \), it signifies looking for all points below this line. The inequality \( y < 7 \) does not include the points on the line itself, only those in the region beneath it.

Here鈥檚 what you can keep in mind:
* The line is horizontal, meaning it affects only the y-coordinate of points.
* Any y-value smaller than 7 satisfies the inequality.

Together with the other inequalities, this line constraint plays a crucial role: it helps trim down the reachable area from above and defines part of the solution area in combination with the parabola and circle regions.

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Most popular questions from this chapter

The average of self-reported spending "yesterday" for high-income consumers and middle-/low-income consumers was 93.50 dollars in September 2012 . High- income consumers spend 65 dollars more than middle-/low-income consumers. (Source: www.marketingcharts.com) (a) Write a system of equations whose solution gives the self-reported spending for each income group. Let \(x\) be the spending by high-income consumers and \(y\) be the spending by middle-/low-income consumers. (b) Solve the system. (c) Interpret the solution.

From 2010 to \(2012,\) the average selling price of tablets decreased by \(30 \% .\) This percent reduction amounted in a decrease of 195 dollars. Find the average selling price of tablets in 2010 and in 2012.

Use the shading capabilities of your graphing calculator to graph each inequality or system of inequalities.$$\begin{aligned} &x+y \geq 2\\\ &x+y \leq 6 \end{aligned}$$

Given a square matrix \(A^{-1}\), find matrix \(A\). $$A^{-1}=\left[\begin{array}{rrr} \frac{2}{3} & -\frac{1}{3} & 0 \\ \frac{1}{3} & -\frac{5}{3} & 1 \\ \frac{1}{3} & \frac{1}{3} & 0 \end{array}\right]$$

As the price of a product increases, businesses usually increase the quantity manufactured. However, as the price increases, consumer demand-or the quantity of the product purchased by consumers-usually decreases. The price we see in the market place occurs when the quantity supplied and the quantity demanded are equal. This price is called the equilibrium price and this demand is called the equilibrium demand. (Refer to Exercise 92 .) Suppose that supply is related to price by \(p=\frac{1}{10} q\) and that demand is related to price by \(p=15-\frac{2}{3} q,\) where \(p\) is price in dollars and \(q\) is the quantity supplied in units. (a) Determine the price at which 15 units would be supplied. Determine the price at which 15 units would be demanded. (b) Determine the equilibrium price at which the quantity supplied and quantity demanded are equal. What is the demand at this price?
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