2.3 Models and applications (Page 4/9)

Algebra and trigonometry 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

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Solving a volume problem

Find the dimensions of a shipping box given that the length is twice the width, the height is $8$ inches, and the volume is 1,600 in. ³ .

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

\begin{matrix} V & = & L W H \\ 1, 600 & = & (2 W) W (8) \\ 1, 600 & = & 16 W^{2} \\ 100 & = & W^{2} \\ 10 & = & W \end{matrix}

The dimensions are $L = 20$ in., $W = 10$ in., and $H = 8$ in.

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Media

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 $d = r t$ formula. See [link] .
Many geometry problems are solved using the perimeter formula $P = 2 L + 2 W,$ the area formula $A = L W,$ or the volume formula $V = L W H .$ See [link] , [link] , and [link] .

Section exercises

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.

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

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$5 (n + 3) = 2 n$

If the total amount of money you had to invest was $2,000 and you deposit $x$ amount in one investment, how can you represent the remaining amount?

$2, 000 - x$

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If a man sawed a 10-ft board into two sections and one section was $n$ ft long, how long would the other section be in terms of $n$ ?

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If Bill was traveling $v$ mi/h, how would you represent Daemon’s speed if he was traveling 10 mi/h faster?

$v + 10$

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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?

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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: $23;$ Beth: $46$

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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?

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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.

Questions & Answers

Cos45/sec30+cosec30=

dinesh Reply

Cos 45 = 1/ √ 2 sec 30 = 2/√3 cosec 30 = 2. =1/√2 / 2/√3+2 =1/√2/2+2√3/√3 =1/√2*√3/2+2√3 =√3/√2(2+2√3) =√3/2√2+2√6 --------- (1) =√3 (2√6-2√2)/((2√6)+2√2))(2√6-2√2) =2√3(√6-√2)/(2√6)²-(2√2)² =2√3(√6-√2)/24-8 =2√3(√6-√2)/16 =√18-√16/8 =3√2-√6/8 ----------(2)

exercise 1.2 solution b....isnt it lacking

Miiro Reply

I dnt get dis work well

john Reply

what is one-to-one function

Iwori Reply

what is the procedure in solving quadratic equetion at least 6?

Qhadz Reply

Almighty formula or by factorization...or by graphical analysis

Damian

I need to learn this trigonometry from A level.. can anyone help here?

wisdom Reply

yes am hia

Miiro

tanh2x =2tanhx/1+tanh^2x

Gautam Reply

cos(a+b)+cos(a-b)/sin(a+b)-sin(a-b)=cotb ... pls some one should help me with this..thanks in anticipation

favour Reply

f(x)=x/x+2 given g(x)=1+2x/1-x show that gf(x)=1+2x/3

Ken Reply

proof

AUSTINE

sebd me some questions about anything ill solve for yall

Manifoldee Reply

cos(a+b)+cos(a-b)/sin(a+b)-sin(a-b)= cotb

favour

how to solve x²=2x+8 factorization?

Kristof Reply

x=2x+8 x-2x=2x+8-2x x-2x=8 -x=8 -x/-1=8/-1 x=-8 prove: if x=-8 -8=2(-8)+8 -8=-16+8 -8=-8 (PROVEN)

Manifoldee

x=2x+8

Manifoldee

×=2x-8 minus both sides by 2x

Manifoldee

so, x-2x=2x+8-2x

Manifoldee

then cancel out 2x and -2x, cuz 2x-2x is obviously zero

Manifoldee

so it would be like this: x-2x=8

Manifoldee

then we all know that beside the variable is a number (1): (1)x-2x=8

Manifoldee

so we will going to minus that 1-2=-1

Manifoldee

so it would be -x=8

Manifoldee

so next step is to cancel out negative number beside x so we get positive x

Manifoldee

so by doing it you need to divide both side by -1 so it would be like this: (-1x/-1)=(8/-1)

Manifoldee

so -1/-1=1

Manifoldee

so x=-8

Manifoldee

SO THE ANSWER IS X=-8

Manifoldee

so we should prove it

Manifoldee

x=2x+8 x-2x=8 -x=8 x=-8 by mantu from India

mantu

lol i just saw its x²

Manifoldee

x²=2x-8 x²-2x=8 -x²=8 x²=-8 square root(x²)=square root(-8) x=sq. root(-8)

Manifoldee

I mean x²=2x+8 by factorization method

Kristof

I think x=-2 or x=4

Kristof

x= 2x+8 ×=8-2x - 2x + x = 8 - x = 8 both sides divided - 1 -×/-1 = 8/-1 × = - 8 //// from somalia

Mohamed

i am in

Cliff

1KI POWER 1/3 PLEASE SOLUTIONS

Prashant Reply

hii

Amit

how are you

Dorbor

well

Biswajit

can u tell me concepts

Gaurav

Find the possible value of 8.5 using moivre's theorem

Reuben Reply

which of these functions is not uniformly cintinuous on (0, 1)? sinx

Pooja Reply

helo

Akash

hlo

Akash

Hello

Hudheifa

which of these functions is not uniformly continuous on 0,1

Basant Reply

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Source: OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6

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