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Chapter 5: Problem 77

Use a graphing utility to approximate the solution(s) to the system of equations. Round the coordinates to 3 decimal places. $$ \begin{array}{l} x^{2}+y^{2}=40 \\ y=-x^{2}+8.5 \end{array} $$

Short Answer

Expert verified
The approximate solutions are at \((\pm 2.438, 2.978)\) and \((\pm 2.438, -2.978)\).

Step by step solution

01

- Understand the equations

The system of equations contains two equations: 1. Circle equation: \[ x^2 + y^2 = 40 \]2. Parabola equation: \[ y = -x^2 + 8.5 \]
02

- Graph the Circle

Using a graphing utility, plot the circle equation \[ x^2 + y^2 = 40 \]. This represents a circle with center at the origin (0, 0) and radius \( \sqrt{40} \).
03

- Graph the Parabola

Using the same graphing utility, plot the parabola equation \[ y = -x^2 + 8.5 \]. This parabola opens downward with the vertex at (0, 8.5).
04

- Identify Intersection Points

After plotting both graphs, identify the points where the circle and the parabola intersect. These intersections are the solutions to the system of equations.
05

- Approximate Intersection Coordinates

Use the graphing utility to find the exact coordinates of the intersection points and round them to three decimal places.

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Key Concepts

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

system of equations
In mathematics, a system of equations is a collection of two or more equations with the same set of unknowns. In our example, we have two equations to solve simultaneously:
  • The circle equation: \(x^2 + y^2 = 40\)
  • The parabola equation: \(y = -x^2 + 8.5\)
Solving a system of equations means finding all sets of values for the variables that simultaneously satisfy all the equations in the system. For this specific problem, the unknowns are the coordinates \(x\) and \(y\) where both the circle and the parabola equations are true at the same time.
circle equation
A circle equation represents all the points \((x, y)\) that are at a fixed distance (radius) from a center point. Our circle equation is:
\[x^2 + y^2 = 40\]
This describes a circle with center at the origin \((0, 0)\) and radius \(\sqrt{40}\) which simplifies to \(2\sqrt{10}\) or approximately 6.325.
Use the graphing utility to plot this circle. All points on the circle have this radius from the center.
parabola equation
A parabola equation represents a symmetrical open curve. The general form of a parabola equation is \(y = ax^2 + bx + c\). In our case, the parabola equation is:
\[y = -x^2 + 8.5\]
This specifies a parabola that opens downward because the coefficient of \(x^2\) is negative. The vertex of this parabola is at \((0, 8.5)\).
Plot this parabola on the same graphing utility as the circle. You'll see that it creates a curved shape that can intersect the circle at certain points.
intersection points
Intersection points are where the graphs of the equations in a system meet. These points satisfy all equations in the system. For our circle and parabola, we look for points that are on both shapes.
Using the graphing utility, plot both graphs:
  • Circle: \(x^2 + y^2 = 40\)
  • Parabola: \(y = -x^2 + 8.5\)
Identify where the two graphs intersect.
These intersection points will give you the solutions to the system. Approximate the coordinates of these points using the graphing tool and round them to three decimal places.
The resulting points are your final answer.

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Most popular questions from this chapter

A manufacturer produces two models of a gas grill. Grill A requires 1 hr for assembly and \(0.4 \mathrm{hr}\) for packaging. Grill \(B\) requires 1.2 hr for assembly and 0.6 hr for packaging. The production information and profit for each grill are given in the table. (See Example 4\()\) $$ \begin{array}{|l|c|c|c|} \hline & \text { Assembly } & \text { Packaging } & \text { Profit } \\ \hline \text { Grill A } & 1 \mathrm{hr} & 0.4 \mathrm{hr} & \$ 90 \\ \hline \text { Grill B } & 1.2 \mathrm{hr} & 0.6 \mathrm{hr} & \$ 120 \\ \hline \end{array} $$ The manufacturer has \(1200 \mathrm{hr}\) of labor available for assembly and \(540 \mathrm{hr}\) of labor available for packaging. a. Determine the number of grill A units and the number of grill B units that should be produced to maximize profit assuming that all grills will be sold. b. What is the maximum profit under these constraints? c. If the profit on grill A units is $$\$ 110$$ and the profit on grill \(\underline{B}\) units is unchanged, how many of each type of grill unit should the manufacturer produce to maximize profit?

A furniture manufacturer builds tables. The cost for materials and labor to build a kitchen table is \(\$ 240\) and the profit is \(\$ 160 .\) The cost to build a dining room table is \(\$ 320\) and the profit is \(\$ 240\). (See Examples \(2-3)\) Let \(x\) represent the number of kitchen tables produced per month. Let \(y\) represent the number of dining room tables produced per month. a. Write an objective function representing the monthly profit for producing and selling \(x\) kitchen tables and \(y\) dining room tables. b. The manufacturing process is subject to the following constraints. Write a system of inequalities representing the constraints. \- The number of each type of table cannot be negative. \- Due to labor and equipment restrictions, the company can build at most 120 kitchen tables. \- The company can build at most 90 dining room tables. \- The company does not want to exceed a monthly cost of \(\$ 48,000\). c. Graph the system of inequalities represented by the constraints. d. Find the vertices of the feasible region. e. Test the objective function at each vertex. f. How many kitchen tables and how many dining room tables should be produced to maximize profit? (Assume that all tables produced will be sold.) g. What is the maximum profit?

An investment grows exponentially under continuous compounding. After 2 yr, the amount in the account is \$7328.70. After 5 yr, the amount in the account is \$8774.10. Use the model \(A(t)=P e^{r t}\) to a. Find the interest rate \(r\). Round to the nearest percent. b. Find the original principal \(P\). Round to the nearest dollar. c. Determine the amount of time required for the account to reach a value of \(\$ 15,000 .\) Round to the nearest year.

Solve the system. $$ \begin{array}{l} 2^{x}+2^{y}=6 \\ 4^{x}-2^{y}=14 \end{array} $$

Write a system of inequalities that represents the points inside the triangle with vertices \((-4,-4),(1,1),\) and (5,-1).
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