what is the equation.?

Question

The perimeter of a rectangle is equal to its area. The height is 3 more than the base. What are the dimensions of the rectangle? the area?

Answer 1

You need to determine base equations first.
let x=the base of the rectangle. therefore x+3=the height of the rectangle.

The perimeter= 2(height+base)
2(x+3+x)
=2(2x+3)
=4x+6

The area=baseXheight
=(x+3)(x)
=x^2+3x

If the two are equal, you set them equal to eachother.

x^2+3x=4x+6
x^2-x+6=0
(x+2)(x-3)
therefore x= -2, 3
since you cant have a negative length, x=3

so the base of the rectangle is 3, and the height=x+3=6

so. the perimeter is 2(base+height)=2(9)=18
the area=baseXheight=3X6=18.

so the perimeter/area are 18.

Answer 2

2(B+H)= P
P=BH
H=3B

so 2(B+H)=BH

but H=3B

so 2(4B)=3B^2

or 8=3B

B=8/3

But H=3B

so H=8

check
2(8+8/3)= 64/3 Perimeter

and 8x8/3 =64/3 Area

and 3 x 8/3 =8 is H 3x the base ? yes

all correct

Answer 3

base x, height x+3
perimeter 2(x + x+3) = 4x + 6
area x(x+3) = x² + 3x

so x² + 3x = 4x + 6
x² - x - 6 = 0
(x - 3)(x + 2) = 0
x = 3, discard x = -2
base x = 3, height x+3 = 6

Answer 4

HW = 2H + 2W
and
H = W + 3
so
(W + 3)W = 2(W + 3) + 2W
W^2 + 3W = 2W + 6 + 2W
W^2 - W - 6= 0
(W-3)(W+2) = 0
Width is 3 and Height is 6
and the area is 18.

Answer 5

perimeter = p = 2a + 2b

area = ab

b =3a

2a +2b = 2a +6a =8a
area = ab = 3a^2
8a =3a^2
8a -3a^2 =0
a(8 -3a) =0
a = 8/3
b = 3a = 8

rectangle has dimensions 8 and 8/3

verification
P = 2(a+b) = 2(8/3+ 8 ) = 64/3
area = ab =( 8/3) (8) = 64/3

Answer 6

P of rectangle (x by 3x):
2x+6x=8x

A of rectangle (x by 3x):
(3x)(x) = 3x^2

P=A therefore:
8x=3x^2

So:
3x^2 - 8x = 0
x(3x-8) = 0
so set 3x-8 equal to zero and you get:
x= (8/3)

Answer 7

L = 3, H = 6
Perimeter = 18
Area = 18
-----------------------
2L + 2H = HL
H = 3 + L
------------------------
2(L+H) = LH --> plug in H = 3+L
2(L+L+3) = L(3+L)
2L + 2L + 6 = 3L + L^2
L^2 - L - 6 = 0
(L-3) * (L+2) = 0
L = 3 or L = -2 (not possible)

Thus L = 3 , H = 6