Question

A rectangle has an area of `30` `cm^2`. It has a width of `5+2x` `cm` and a length of `2x+4` `cm`. What is the length and width?

Answer 1

The width is `5` cm and the length is `6` cm

Explanation:

If you know the length `l` and width `w` of a rectangle, the area `A` is given by their product. So, you have

`A = w*l \implies 30 = (5+2x)(2x+4)`

(we just translated the known values/expressions for area, width and length).

We can expand the width-length product, multiplying each term in the first parenthesis times each term in the second parenthesis:

`(5+2x)(2x+4) = 5*2x+5*4+2x*2x+2x*4`

`= 10x+20+4x^2+8x`

`= 4x^2+18x+20`

We want this quantity to equal `30`, so we have

`4x^2+18x+20 = 30`

subtracting `30` from both sides...

`4x^2+18x-10 = 0`

You can solve this equation with the usual quadratic formula to get

`x_{1,2} = \frac{-18\pm\sqrt{324+160}}{8}=\frac{-18\pm\sqrt{484}}{8} = \frac{-18\pm 22}{8}`

If we choose the plus sign, we have

`\frac{-18 + 22}{8} = \frac{4}{8} = \frac{1}{2}`

If we choose the minus sign, we have

`\frac{-18 - 22}{8} = \frac{-40}{8} = -5`

Given those values for `x`, we can try to deduce the width and length: considering the first solution `x = 1/2` we have

`w = 5+2x \implies w = 6` cm
`h = 4+2x \implies h = 5` cm

Considering the second solution `x = -5` we have

`w = 5+2x \implies w = -5` cm
`h = 4+2x \implies h = -6` cm

This solution makes algebraically sense, since the area would be `(-5cm)(-6cm) = 30cm^2`, but we can't have sides of a rectangle with negative measures, so we must reject these solutions.

Answer 2

`color(blue)("Length"=6cm)`

`color(blue)("Width"=5cm)`

Explanation:

The area of a rectangle is given by:

`"Area"="length"xx"width"`

We are given `"Area"=30cm^2`

Therefore:

`"Area"=(2x+4)*(5+2x)=30`

`\ \ \ \ \ \ \ \ \ \=4x^2+18x+20=30`

`\ \ \ \ \ \ \ \ \ \=4x^2+18x-10=0`

Factor:

`(2x+10)(2x-1)=0=>x=-5 and x=1/2`

We now check these, since we can't have a negative side to the rectangle:

`x=-5`

`(2(-5)+4)=-6`

We discard this solution.

`x=1/2`

`(2(1/2)+4)=5`

`(5+2(1/2))=6`

So:

`"Length"=6cm`

`"Width"=5cm`