Your selection 1. Which ball did you select? What do you know about the base of the box? (2 points: 1 point for the selection, 1 point for the information)
Modeling your box
I selected a volleyball. The box has a square base.

2. Here is some more information about making the boxes:
• SportsBounceCo uses flat sheets of cardboard to make boxes.
• The company uses square sheets for volleyball boxes and rectangular sheets for football boxes. • The boxes will have a clear plastic top, so customers can see the product.
• The height of the box is always 1 inch greater than the width of the ball.
• To assemble the box, corners are cut out of each sheet and the edges are taped together.

Draw and label the necessary dimensions for your cardboard sheet. Use x for the width and x + 1 for the height. (4 points: 2 points for the drawing, 2 points for correct labels)

3. Now use your drawing to write an equation for the area of the entire sheet of cardboard. First write the equation as the product of two binomials, and then as a simplified trinomial. You can the tile tool to check your work by moving x-tiles and unit tiles to the vertical and horizontal bars in the tool. (4 points: 2 points for the product of binomials, 2 points for trinomial form)

4. Next write an equation for the surface area of the box (after the sheet has been folded). Don't include a top for the box; the top will be added later. (2 points: 2 points for the correct equation)
5. Fill in the table below to calculate the amount of material wasted in producing each size of box. Show your work. (6 points: 2 points for each different width)
Ball diameter
Area of full sheet of cardboard
Surface area of box
Area removed from corners
8 in
9 in
10 in

Ball diameter
Area of full sheet of cardboard
Surface area of box
Area removed from corners
4 in
5 in
6 in
A fraction of a problem
6. SportsBounceCo makes only boxes that have sides that are measured in whole inches, like the boxes you have been describing so far. Is it possible for them to produce a box that has a surface area that is not a whole number? How do you know?
(2 points: 1 point for a correct response, 1 point for an explanation)

answer: learning objective(s)

·         describe the creation and use of systems of equations.

·         graph a system of linear equations on the coordinate plane and identify its solution.

introduction

sometimes graphing a single linear equation is all it takes to solve a mathematical problem. other times, one line just doesn’t do it, and a second equation is needed to model the situation. this is often the case when a problem involves two variables. solving these kinds of problems requires working with a system of equations, which is a set of two or more equations containing the same unknowns.

let’s take a look at systems of equations, and see what the graphs of individual equations within a system reveal about the mathematical relationship of the variables.

systems of equations

a system of equations contains two or more linear equations that share two or more unknowns. to find a solution for a system of equations, we must find a value (or range of values) that is true for all equations in the system.

the graphs of equations within a system can tell us how many solutions exist for that system. look at the images below. each show two lines that make up a system of equations (in the graph on the right the two lines are superimposed and look like a single line). how many points in common does each system of lines reveal?

one solution

no solutions

infinite solutions

if the graphs of the equations intersect, then there is one solution that is true for both equations.

if the graphs of the equations do not intersect (for example, if they are parallel), then there are no solutions that are true for both equations.

if the graphs of the equations are the same, then there are an infinite number of solutions that are true for both equations.

remember, the graph of a line represents every point that is a possible solution for the equation of that line. so when the graphs of two equations cross, the point of intersection lies on both lines, meaning that it is a possible solution for both equations. when the graphs of two equations never touch, there are no shared points and there are no possible solutions for the system. when the graphs of two equations lie on top of one another, they share all their points and every one is a possible solution.

graphing as a solution method

graphing equations in order to identify a specific point of intersection is usually not a precise way to solve systems because it is often difficult to see exactly where two lines intersect (unless you are using a computer-based graphing program that allows you to zoom in on a point). however, the graph of a system of equations can still give a good idea of what type of solution, if any, exists. let’s graph a system, and see how it works.

graph the system y = 3x and x + 2y = 4. how many solutions does this system have?

a graph of the two lines y = 3x and x + 2y = 4 shows us that the lines intersect, meaning that there is a single (x, y) value that satisfies both equations. looking at the graph does not tell us exactly where that point is, but we don’t need to know that information here, because the question only asks for the number of solutions.

so a system made of two intersecting lines has one solution. now let’s look at a different situation:

how many solutions exist for the system y -0.5x = 7 and 2y = x − 3?

plotting both equations, it looks like there is no solution—the lines are parallel. to check this finding, we can compare the slopes of the equations. to make that easy, we’ll rearrange the equations into slope-intercept form, or y = mx + b.   that gives us the equations y = 0.5x + 7 and y = 0.5x − 1.5.   yes, the slope of both lines is 0.5, which means the lines really are parallel. they never intersect, so there is no point that lies on both lines, and no solution to the system.

step-by-step explanation:

3x + 18 = 147°

step-by-step explanation:

3x + 18 = 147°, because the value of the central angle is equal to the value of the arc formed by it.

i hope i you.