Question

Area of Rectangle: A = length*width

One side of a rectangle is twelve kilometers longer than three times another side of the rectangle. Find the
sides if we also know that the area of the rectangle is 135 km².
shorter side:
longer side:

Answer 1

Short side: 5 km

Long side: 27 km

Explanation:

let x = length of shorter side

`(3x + 12)(x)= 135` This is the equation

`3x^2+12x=135` use distributive property to multiply out

`3x^2+12x-135=0` subtract 135 from both sides to get in position to factor

`3(x^2+4x-45)` factor out a 3

`3(x-5)(x-9)` factor the resulting expression

`x=5,9` two answers (we want one so see which one works)

Check both values of x

`5*3+12=27`

`27*5=135`

Therefore, we know the shorter side is 5 km and the longer side is 27 km.

At this point, we do not even need to check x=9

Answer 2

shorter side: 5 km

longer side: 27 km

Explanation:

So you're already given the formula for the area of a rectangle, the next thing to do is assign length and width in terms of some unknown variable, which for convenience, I'll just say is x. So let's say one of the sides is x, the one that's the shorter side, since the longer side can be related to the shorted side, which is x. The longer side is 12km longer than 3 times the other side, so since the other side is x, it's length is (3x+12). So we simply multiply this by x, to get the area which is 135

"Known" Values: one side=x, other side=3x+12, area=135 km^2

`135=x(3x+12)`

Distribute the x

`135=3x^2+12x`

Subtract 135 from both sides

`0=3x^2+12x-135`

Now since the equation is equal to zero, we can use the quadratic equation. In this case a=3, b=12, and c=-135

`x=(-12\pm√(12^2-4(3)(-135)))/(2(3))`

Simplify denominator and multiply inside of the radical

`x=(-12\pm√(144+1,620))/(6)`

If you're confused where the negative went in the radical, it canceled out when multiply -4 by -135. Now add the stuff in the radical

`x=(-12\pm√(1,764))/(6)`

Simplify the radical

`x=(-12\pm 42)/(6)`

Now you might think there are two solutions to this equation, but the thing is that one of the sides simply cannot have a "negative length", because in this context, x must be positive. So it's easy to see you have to use the positive sign, since using the negative sign would result in -12-42 over 6 which is certainly negative. So we can ignore that solution as it doesn't mean anything in this context. We only need the positive solution

`x=(-12+42)/(6)`

Add values

`x=(30)/(6)`

Divide

`x=5`

Since the shorter side was 5, that's the solution to the shorter side, now to find the longer side we plug in 5 into the equation:
`3x+12` which becomes 3(5) + 12 = 15+12 = 27. This means the two lengths are 5 and 27