Algebra: Quadratic Word Problem?

Question

Abbie is an interior designer who has been asked to find an oriental rug for a new corporate office. As a rule, the rug should cover 1/2 of the total floor area with a uniform width surrouding the rug. If the dimensions of the room are 12 feet by 16 feet, find the dimensions of the rug that Abbie should look for.


I'm having so much trouble with this problem, I did it like 10x times and kept on getting different answers! Please explain it step by step if possible thanks!

Answer 1

we can let the uniformed width be x
then one dimension of the rug would have to be 12-2x (taking the width from both ends)
and the other dimension of the rug would have to be 16-2x (taking the width from both ends)
product of the dimensions = area
(remember area of rug is only half of the area original 12*16 = 1/2 (192)= 96)
we get
(12-2x)(16-2x) = 1/2 (192)
(12-2x)(16-2x) = 96
192 - 56x +4x² = 96
4x² - 56x + 192 - 96 = 0
4x² - 56x + 96 = 0 factorizing common factor 4 gives
4(x² - 14x + 24) = 0
since 4 is not =0 then x² - 14x + 24 = 0 factorizing completely gives (x-12)(x-2) = 0
so x-12 = 0 or x-2=0
x = 12 or 2
the width cannot be 12 so it must be 2
so the dimensions are 12-2(2) = 12 - 4 = 8
and 16 - 2(2) = 16 - 4 = 12
dimensions of the rug is 8 x 12

hope that helped

Answer 2

Looking at the numbers one coudl giuve you the answer(8x12) without writting anything down, and because i can see that the answers are nice rounded whol enumber i will walk you through a foolprooof (but long) way to know you are correct in your answer:

there are two rules to follow (form the given info):
#1 ... rug should cover half the area
AREA(room) = 12*16 = 192'
AREA(rug) = 192/2 = 96'

#2 ... there has to be an even space around all edges of the carpet .. meaning the rug will be 12-n and 16-n

Now we know that the L*W = 96 and have the following options: ( i told you it woudl be long!)
12 -1 = 11 ... & ... 16 - 1 = 15
... L*W = 11*15 = 165 .. too big

12 -2= 10 ... & ... 16 -2 = 14
... L*W = 10*14 = 140 .. too big

12 -3= 9 ... & ... 16 -3 = 13
... L*W = 9*13 = 117 .. too big

12 -4= 8 ... & ... 16 -4 = 12
... L*W = 8*12 = 96' ... perfect!

ANS: 8*12

Answer 3

Let the uniform width be x ft

then length of the rug = 16 - 2x (leaving x ft on either side)

the widh of the rug = 12 - 2x

area of rug = (16 - 2x)(12-2x) = 192 - 32x - 24x + 4x^2

=>4x^2 - 56x + 192

since rug's area is 1/2 of the area of the floor

4x^2 - 56x + 192 = (16*12)/2 = 96

4x^2 - 56x + 96 = 0

divide by 4

x^2 - 14x + 24 = 0

x^2 - 12x - 2x + 24 = 0

x(x - 12) - 2(x - 12) = 0

(x -12)(x - 2) = 0

x = 12 or 2

12 is not a valid solution, x = 2ft

the dimensions of rug

length = 16 - 2(2) = 16 - 4 = 12 ft

width = 12 - 2(2) = 12 - 4 = 8 ft

Answer 4

1/2 the total area would be 96 ft^2. Consider a uniform width removed from all four sides of the rectangle, call it x. The length of the new rug is 12-x and the width is 16-x.

Multiply the two and set equal to 96^2. Solve the resuting quadratic from which you'll get two answers for x, one which will be clearly be wrong, the other will be the answer.

Answer 5

let the width of the floor surrounding the rug

(12-2x)(16-2x) = 1/2 12*16 = 96
192 - 56x +4x^2 = 96
4x^2 - 56x + 96 = 0
x^2 - 14x + 24 = 0
(x-12)(x-2) = 0
x = 12 or 2
it cannot be 12 because the floor is only 12x16
so th rug is 8x12

Answer 6

enable the harmful huge type you're searching for be represented by "a" then, you are able to set up an equation depending off the data given: (-a+3)^2=1521 ensure for a (-a+3)= +/- Sqrt(1521) -a= +/- Sqrt(1521) eliminate the harmful verify in the left by multiplying by -a million So, a = -Sqrt(1521)+3 or a =+Sqrt(1521)-3 even if the question asks for a unfavorable actual huge type, so the answer for this question is a= -Sqrt(1521)+3= -36