We know that the area of a rectangle is solved via the equation:
[tex]A=l*w[/tex]
And we are told that the area of the rectangle is 54[tex] m^{2} [/tex], so:
[tex]54 m^{2}=l*w [/tex]
We are told that the length is 3 meters more than twice the width, so we can then set up the equation:
[tex]l=2w+3[/tex], with w representing the width of the rectangle
Let's plug the above equation into the equation representing area:
[tex]54=(2w+3)*w [/tex]
Simplify the right side:
[tex]54=2 w^{2}+3w [/tex]
And then subtract 54 from both sides to set the equation equal to 0 for factoring purposes:
[tex]0=2 w^{2}+3w-54 [/tex]; then factor:
[tex]0=(2w-9)(w+6)[/tex]
Set each term equal to 0, and solve for w. We get the answers:
[tex] \frac{9}{2} [/tex] and [tex]-6[/tex]
Since the width cannot be a negative number, we take the positive value as the true width. Now we know that the width is [tex] \frac{9}{2} [/tex] meters.
Let's plug this value into the equation for the length:
[tex]l=2w+3[/tex]
[tex]l=2( \frac{9}{2})+3 [/tex]
[tex]l=9+3=12[/tex]
So now we know that the length is 12 meters.
Overall, the length of the rectangle is 12 meters and the width of the rectangle is [tex] \frac{9}{2} [/tex] meters.
