Question

the width of a rectangle is two less than twice the length. the area of the rectangle is equal to eight inches more than double the perimeter. what is the width in inches

Answer 1

The value of width is 12 inches

Explanation:

Through the statement we can propose the following equations

Frist, "the width of a rectangle is two less than twice the length" would be:

let W for width and L for length

`W=2L-2`

Second, "the area of the rectangle is equal to eight inches more than double the perimeter." woluld be:

Let P for perimeter and A for area:

`A=8+2P`

We know that the perimeter is equal to:

`P=2W+2L`

Replacing:

`A=8+2\cdot(2W+2L)`

And by formula we know that the area is equal to:

`A=W\cdot L`

Replacing:

`W\cdot L=8+2\cdot(2W+2L)`

Now, we replace what the width equates to as a function of length and solve for the length:

`\begin{gathered} (2L-2)\cdot L=8+2\cdot(2\cdot(2L-2)+2L) \\ 2L^2-2L=8+2\cdot(4L-4+2L) \\ 2L^2-2L=8+12L-8 \\ 2L^2-2L-12L=0 \\ 2L^2-14L=0 \\ 2L\cdot(L-7)=0 \\ 2L=0\rightarrow L=0 \\ L-7=0\rightarrow L=7 \end{gathered}`

Now, for W, replacing:

`\begin{gathered} W=2\cdot7-2=14-2=12 \\ W=12 \end{gathered}`