Question

The lenght is (x-4) and the width is (x+6). Find the perimeter if the Area is 144.

A. 50
B.52
c. 48
d. 54

Answer 1

The formula for the area of a rectangle is A = length × width. We are given that the area is 144, so we can write:

144 = (x-4)(x+6)

Expanding the right side, we get:

144 = x^2 + 2x - 24

Adding 24 to both sides, we get:

168 = x^2 + 2x

Rearranging, we get:

x^2 + 2x - 168 = 0

Now we can factor the left side:

(x + 14)(x - 12) = 0

So we have two possible solutions: x = -14 and x = 12. However, since the length and width of a rectangle cannot be negative, we must choose x = 12.

Therefore, the length is (12-4) = 8 and the width is (12+6) = 18. The perimeter is:

2(length + width) = 2(8 + 18) = 52

So the answer is (B) 52.

Answer 2

B

Explanation:

Given:

l (length) = (x - 4)

w (width) = (x + 6)

A (area) = 144

Find: P (perimeter) - ?

We can write an equation according to the given information:

A = w × l

`(x - 4)(x + 6) = 144`

Expand the brackets - multiply every term inside the bracket by the term on the outside:

`{x}^(2) + 6x - 4x - 24 = 144`

Collect like-terms (don't forget, when moving terms to the other side, make sure to change their sign into the opposite of the previous one):

`{x}^(2) - 2x- 24 - 168 = 0`

`{x}^(2) + 2x - 168= 0`

a = 1; b = 2; c = (-168)

Solve this quadratic equation:

`d = {b}^(2) - 4ac = {2}^(2) - 4 * 1 * ( - 168) = 4 + 672 = 676 > 0`

`x1 = ( - b - √(d) )/(2a) = ( - 2 - 26)/(2 * 1) = ( - 28)/(2) = - 14`

x must be a natural number, since side lengths cannot have negative units

`x2 = ( - b + √(d) )/(2a) = ( - 2 + 26)/(2 * 1) = (24)/(2) = 12`

P = 2l + 2w

P = 2( x - 4 ) + 2( x + 6)

Replace x with its new value (x2):

P = 2(12-4) + 2(12+6) = 2*8 + 2*18 = 16 + 36 = 52