two-variable inequality

 

Two-Variable Inequality

 

Read the following instructions in order to complete this assignment, and review the example of how to complete the math required for this assignment: 

Read problem 46 on page 240 of Elementary and Intermediate Algebra.

 

Assign a variable to each type of rocker Ozark Furniture makes.

 

Write a linear inequality which incorporates the given information of total board feet and the board feet required for each type of rocker.

 

On scratch paper draw a graph of the inequality so that you have this visual to go by as you discuss the graph in your writing.  A scanned copy of this graph may be attached with your essay, but is not required.

 

Write a two to three page paper that is formatted in APA style and according to the Math Writing Guide. Format your math work as shown in the Instructor Guidance and be concise in your reasoning. In the body of your essay, please make sure to include:

 

Your solution to the above problem, making sure to include all mathematical work.

 

A discussion on what this graph looks like. Include information about the intercepts, the type of line needed, direction of the line, and region(s) shaded to fulfill the inequality. Any details which are pertinent to know about the graph should be mentioned.

 

An application of the findings in this graph. For example, pick a point in the shaded area and give its coordinates, and then discuss what those numbers mean in terms of rockers and board feet of lumber. Pick a point outside of the shaded area and do the same thing. Pick a point right on the line and discuss the same details. Be specific.

 

An evaluation of the following: A chain furniture store faxes an order for 175 modern rocking chairs and 125 classic rocking chairs. Will Ozark Furniture be able to fill this order with the current lumber on hand? If yes, how much lumber will they have left? If no, how much more lumber would they need to fill the order?

 

INSTRUCTORGUIDANCE EXAMPLE: Week Three Assignment

2-Variable Inequality

Here is an example of a problem very similar to the one in the Week Three Assignment:

Catskills Hammock Company can obtain at most 2000 yards of striped canvas for making

its full size and chair size hammocks. A full size hammock requires 10 yards of canvas

and the chair size requires 5 yards of canvas. Write an inequality that limits the number

of striped hammocks of each type which can be made.

(b) First I must define what variables I will be using in my inequality.

Let f = the number of full size hammocks

Let c = the number of chair size hammocks

Since each full size hammock requires 10 yards of canvas I will use 10f, and since each

chair hammock requires 5 yards of canvas I will use 5c. The total amount of canvas

which can be used is limited to 2000 yards because that is all they can get. Together my

inequality will look like this:

10f +

5c ≤ 2000

(d) If we call f the independent variable (on the horizontal axis) and c the dependent

variable (on the vertical axis) then we can graph the equation using the intercepts.

The f-intercept is found when c = 0:

10f ≤ 2000

f ≤ 200 The f-intercept is (200, 0).

The c-intercept is found when f = 0:

5c ≤ 2000

c ≤ 400 The c-intercept is (0, 400).

Because this is a “less than or equal to” inequality the line will be solid, sloping

downward as it moves from left to right. The region of the graph which is relevant to this

problem is restricted to the first quadrant, so the shaded section is from the line towards

the origin and stops at the two axes.

(e) Consider the point (105, 175) on my graph. It is inside the shaded area which means

the company could fill an order of 105 full size hammocks and 175 chair hammocks. If

they made up this many items they would use

105(10) + 175(5) = 1925 yards of striped canvas and have 75 yards left over.

Consider the point (150, 125) on the graph. It is outside the shaded area which means the

company could not make up enough of both kinds of hammocks to fill this order. They

would run out of canvas before all of them got made.

150(10) + 125(5) = 2125 yards of canvas required. Cannot fill the order.

Consider the point (75, 250). This point is right on the line and this means the company

could fulfill this order exactly without any canvas left over.

75(10) + 250(5) = 2000 Order is filled but no room for mistakes!

(f) If someone calls and submits an order for 120 full size hammocks and 180 chair

hammocks would the company be able to fill this order?

On the graph one can see that this point (120, 180) is outside of the shaded area so

this tells us that no, the company cannot make enough striped hammocks to fill this order.

By plugging in the numbers for each type of hammock and evaluating we can see how

much short of enough canvas the company is.

120(10) + 180(5) = 1200 + 900 = 2100 yards of canvas needed.

The company is 100 yards of canvas short of being able to fill this order.

[Students should remember this example does not demonstrate APA style, nor the

inclusion of introductory or concluding paragraphs.]