# 2.3 Models and applications  (Page 4/9)

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A game room has a perimeter of 70 ft. The length is five more than twice the width. How many ft 2 of new carpeting should be ordered?

250 ft 2

## Solving a volume problem

Find the dimensions of a shipping box given that the length is twice the width, the height is $\text{\hspace{0.17em}}8\text{\hspace{0.17em}}$ inches, and the volume is 1,600 in. 3 .

The formula for the volume of a box is given as $\text{\hspace{0.17em}}V=LWH,$ the product of length, width, and height. We are given that $\text{\hspace{0.17em}}L=2W,$ and $\text{\hspace{0.17em}}H=8.\text{\hspace{0.17em}}$ The volume is $\text{\hspace{0.17em}}1,600\text{\hspace{0.17em}}$ cubic inches.

$\begin{array}{ccc}\hfill V& =& LWH\hfill \\ \hfill 1,600& =& \left(2W\right)W\left(8\right)\hfill \\ \hfill 1,600& =& 16{W}^{2}\hfill \\ \hfill 100& =& {W}^{2}\hfill \\ \hfill 10& =& W\hfill \end{array}$

The dimensions are $\text{\hspace{0.17em}}L=20\text{\hspace{0.17em}}$ in., $\text{\hspace{0.17em}}W=10\text{\hspace{0.17em}}$ in., and $\text{\hspace{0.17em}}H=8\text{\hspace{0.17em}}$ in.

Access these online resources for additional instruction and practice with models and applications of linear equations.

## Key concepts

• A linear equation can be used to solve for an unknown in a number problem. See [link] .
• Applications can be written as mathematical problems by identifying known quantities and assigning a variable to unknown quantities. See [link] .
• There are many known formulas that can be used to solve applications. Distance problems, for example, are solved using the $\text{\hspace{0.17em}}d=rt\text{\hspace{0.17em}}$ formula. See [link] .
• Many geometry problems are solved using the perimeter formula $\text{\hspace{0.17em}}P=2L+2W,$ the area formula $\text{\hspace{0.17em}}A=LW,$ or the volume formula $\text{\hspace{0.17em}}V=LWH.\text{\hspace{0.17em}}$ See [link] , [link] , and [link] .

## Verbal

To set up a model linear equation to fit real-world applications, what should always be the first step?

Answers may vary. Possible answers: We should define in words what our variable is representing. We should declare the variable. A heading.

Use your own words to describe this equation where n is a number:

$5\left(n+3\right)=2n$

If the total amount of money you had to invest was $2,000 and you deposit $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ amount in one investment, how can you represent the remaining amount? $2,000-x$ If a man sawed a 10-ft board into two sections and one section was $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ ft long, how long would the other section be in terms of $\text{\hspace{0.17em}}n$ ? If Bill was traveling $\text{\hspace{0.17em}}v\text{\hspace{0.17em}}$ mi/h, how would you represent Daemon’s speed if he was traveling 10 mi/h faster? $v+10$ ## Real-world applications For the following exercises, use the information to find a linear algebraic equation model to use to answer the question being asked. Mark and Don are planning to sell each of their marble collections at a garage sale. If Don has 1 more than 3 times the number of marbles Mark has, how many does each boy have to sell if the total number of marbles is 113? Beth and Ann are joking that their combined ages equal Sam’s age. If Beth is twice Ann’s age and Sam is 69 yr old, what are Beth and Ann’s ages? Ann: $\text{\hspace{0.17em}}23;$ Beth: $\text{\hspace{0.17em}}46$ Ben originally filled out 8 more applications than Henry. Then each boy filled out 3 additional applications, bringing the total to 28. How many applications did each boy originally fill out? For the following exercises, use this scenario: Two different telephone carriers offer the following plans that a person is considering. Company A has a monthly fee of$20 and charges of $.05/min for calls. Company B has a monthly fee of$5 and charges \$.10/min for calls.

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