Question

the area of a rectangle is 135 feet we know that one side is 12 feet longer than three times the other side the rectangle has dimensions a and b where a less than or equal to b so a the shorter side is how many feet

Answer 1

This is our rectangle:

We know that the area of a rectangle is it's length times it's width:

$A=b\times a$

The problem says that one side is 12ft longer than 3 times the other side, so if a is the shorter side:

$b=3a+12$

And we know that the area is 135ft². If we replace this and the expression for b into the equation for the area we can clear a and then find b:

$\begin{gathered} 135=b\times a \\ 135=(3a+12)a \\ 135=3a^{2}+12a \\ 0=3a^{2}+12a-135 \end{gathered}$

Note that we got a 2 degree equation. We can solve it this way:

$\begin{gathered} 0=Ax^{2}+Bx+C \\ x=\frac{-B\pm\sqrt[]{B^{2}-4AC}}{2A} \end{gathered}$

In our equation we have that A=3, B=12 and C=135 (and instead of x we have a):

$\begin{gathered} a=\frac{-B\pm\sqrt[]{B^2-4AC}}{2A} \\ a=\frac{-12\pm\sqrt[]{12^2+4\cdot3\cdot135}}{2\cdot3}=\frac{-12\pm\sqrt[]{144^{}+1620}}{6}=\frac{-12\pm\sqrt[]{1764}}{6}=\frac{-12\pm42}{6} \end{gathered}$

One of these results will be negative, but since a is the size of the side of a rectangle we don't want a negative number, so we'll keep the positive result:

$a=\frac{-12+42}{6}=\frac{30}{6}=5$

So we got that side a = 5 feet

Then we find b with the expression:

$b=3a+12=3\cdot5+12=15+12=27$

Side b = 27 feet