Question

The width and the length of a rectangle are consecutive even integers. if the width is decreased by 3 inches, then the area of the resulting rectangle is 24 square inches. what is the area of the original rectangle? 12 square inches 48 square inches 96 square inches

Answer 1

Consecutive even integers = x, x + 2

(x - 3)(x + 2) = 24 Since width x, was decreased by 3.

If we test x = 6, (6 - 3)(6 + 2) = 3*(8) = 24

So to (x - 3)(x + 2) = 24, x = 6 satisfies it.

The original rectangle = x, x + 2 = 6, 6 + 2 = 6, 8

Original area = 6*8 = 48 square inches

Answer 2

The area of the original rectangle is
`48in^(2)`

Explanation:

Let's define the following variables :

`W` : ''Width''

`L` : ''Length''

We know that the area of the original rectangle is :

`Area=(W).(L)`

We need to find the values of this variables in order to calculate the original area of the rectangle.

We know that if the width is decreased by 3 inches, then the area of the resulting rectangle is
`24in^(2)`. We can write the following equation :

`(W-3).(L)=24` (I)

We know that the width and the length are consecutive even integers.

Therefore we have two cases :

`W+2=L` (A)

`W-2=L` (B)

Let's suppose the case (A). So if we replace the equation of the case (A) in (I) we will obtain :

`(W-3).(L)=24`

`(W-3).(W+2)=24`

`W^(2)+2W-3W-6=24`

`W^(2)-W-30=0`

If we use the quadratic formula we will obtain two possibles values for
`W` :

`W_(1)=6` and
`W_(2)=-5`

`W_(2)` is absurd because a length can't be negative. The value
`W_(1)` is possible.

If
`W=6` , then using the equation of the case (A), we obtain that

`W+2=L\\6+2=L\\L=8`

The pair :

`W=6\\L=8`

is a possible solution for the problem. If we use the equation of the case (B) in (I) we will obtain the following expression :

`W^(2)-5W-18=0`

If we use the quadratic formula in this equation we will obtain that

`W_(3)=(5+√(97))/(2)\\W_(4)=(5-√(97))/(2)`

This expressions are absurd because
`W` must be an even integer number.

Finally, the solution
`W=6` ,
`L=8` is the only correct solution.

Calculating the area of the original rectangle :

`Area=(W).(L)=(6).(8)=48`

The area of the original rectangle is
`48in^(2)`