What's the length of each side of a rectangle?

Question

We already know that rectangle has segment X and Y.

Area is 108 square units and the perimeter is 42 units.

What's the measure of X and Y

How to find the algebraic answer?

Answer 1

the sides are 9 and 12 units long. Here is the algebra:
area: a=xy=108
perimiter:p=2x+2y=42
-2y -2y
=(2x=42-2y)/2
=x=(-y+21)
substitute that back into the area formula:
(-y+21)(y)=108
0=-y^2+21y-108
use the quadratic formulat to get the roots 9,12

Answer 2

Perimeter of a Rectangle = 2 (X+Y) = 42 i.e. X+Y = 21 X = 21-Y
Now
Area of a Rectangle = X * Y = 108

(21-Y)*Y=108

Y^2-21Y+108=0

Y^2-12Y-9Y+108=0

Y(Y-12)-9(Y-12)=0

(Y-9) (Y-12) = 0

Y = 9 or 12

If Y = 9 then X=12

and If Y=12 then X=9

Answer 3

Area = (x)(y)
108 = (x)(y)

So, y = 108/x

Substitute y into the perimeter equation:
Perimeter = 2x + 2y
42 = 2x + 2(108/x)

Now solve for x

42 = 2x + 216/x
42x = 2x^2 +216 (multiply everything by x)
0 = 2x^2-42x+216 (quadratic equation)
0 = x^2-21x+108 (divide everything by 2)
0 = (x-12)(x-9)

Therefore, x can be 9 or 12.
When substituted into the area equation, 108=(x)(y)
you find that y is 9 or 12.

So, one side has length 9 and the other has length 12.

Answer 4

well, x = 9 and y = 12... or y = 9 and x = 12

you solve this using a system of equations:

xy = 108 and 2x + 2y = 42

to find a variable, you must solve for one of them.
i used xy = 108 to get y = 108/x
then i substituted 108/x for y in 2x + 2y = 42
i solved that algebraically (when you get to -x^2 + 21x = 108, set the equation equal to 0; make it -x^2 +21x -108 = 0, and solve for x by factoring), and got x = 9 and x = 12. Either would work

then you substitute x (9 or 12) in one of your two equations to get y. I used xy = 108 and got my answer.


ncaafan2 did it incorrectly

Answer 5

well you would make it X*Y=108
and then 2X+2Y=42 and then use substitution
so y=42/X

2X + 2(108/x) = 42
2X + 108/2x =42
so then 108/2x= 42-2X
and then multiple by 2x, so 216=2x(42-2X)
216=84X-4X^2
so you have the quadratic equation -4X^2 +84X - 216 = 0
so you would have 18 as an answer for X and then if you pluf it back in you would do 108/18 and get 6 as an answer.

So X=18 and Y = 6

Answer 6

area is given by x*y

xy=108

perimeter is 2*(x+y)

2(x+y)=42

x+y=21

x=21-y

substituting this in area eqn

(21-y)y=108

y^2-21y+108=0

factoring this

(y-21)(y-9)=0

y=21 or 9

but y cant be 21 as x =0 in that case

therefore y=9 and x=12