How to find dimensions to thes problems?

Question

1. Find the dimensions of a rectangle whose perimeter is 46 m and whose area is 126 m (squared).

2. Find the dimensions of a rectangle whose perimeter is 42 m and whose area is 104 m (squared)

Answer 1

1.
2a + 2b = 46
a * b = 126

It gives side a = (46 - 2b) / 2 = 23 -b
(23 - b) * b = 126
23b - b^2 -126 = 0
quadratic equ. you find your 2 possible values of b
then a = 23 -b
Keep the solutions where a > 0 & b>0

Answer 2

It's a set of 2 equations.

First, let x be one side length of the rectangle
let y be the other length

the perimeter would be expressed by the equation (A)
2x + 2y = 46
we can simplify this to (B)
x + y = 13

the area would be expressed by (C)
xy = 126

Start by using equation (A) and solve for x:
x + y = 13 : x = 13 - y

Now you know x (in terms of y), so we'll plug that value into equation (C)
xy=126 : (13 - y)(y) = 126 : 13y - y² = 126

Put that into ay² + by + c = 0 form, and you get
y² - 13y + 126 = 0

Using the quadratic formula (see link below), we solve for y and get the values 9 and 14.

Plug those into equation (B) to get the other side length, and you find that the sides are 9 and 14m in length.

Now get to work on the 2nd one!

Hope this helped!

Answer 3

1)assume that length = a meter and
width = b meter
2 * (a + b) = 46 which means that a + b = 23; so,
b = 23 - a.
a * b = 126
so, a * (23 - a) = 126
23a - a^2 = 126
a^2 - 23a + 126 =0
(a - 14) * (a -9) = 0
a = 14 or 9
So one side is 14m and the other side is 23 - 14 =9m

the second question can be solved in the same way.

Answer 4

l x w = 126 m (1)
2l + 2 w = 46 m (2)

l + w = 23 (divide perimeter eqn by 2)
l = 23 -w (3)
sub (3) in (1)
(23 - w) w = 126
-w^2 + 23w -126 = 0
w^2 - 23w + 126 = 0
(w-14)(w-9) = 0
w = 14 or w = 9
plug answers in (3)
l = 9 or l = 14.

Using the convention l > w, choose:
l = 14, w = 9

Answer 5

xy = 126
2x + 2y = 46
x = 23 - y
y(23 - y) = 126
y² - 23y + 126 = 0
(y - 14)(y - 9) = 0
The dimensions are 9 x 14

xy = 104
2x + 2y = 42
x = 21 - y
y(21 - y) = 104
y² - 21y + 104 = 0
(y - 13)(y - 8) = 0
The dimensions are 8 x 13

Answer 6

2 equations, 2 unknowns:
2*(p+q) = 46
p*q = 126

crank this and then do #2 yourself