The length of a rectangular garden is 12 feet more than the width. If the area of the garden is 189 square feet what are the measurements of the length and width?

Question

algebra 2 and finding out measurements to a word problem

Answer 1

The key here is to look at the formula for area in a rectangle and write an algebraic equation.

If w is the width, and L is the length, then Aera = Width * Length = WL

But the problem gives both area and length as a function of width. Or simply, L = W + 12.

Plug that into the equation, as well as the area = 189 and you get:

W * (W + 12) = 189.

W^2 + 12W = 189 (1)

W^2 + 12W - 189 = 0 (2)

There are a few ways to solve this, either keep it as (1) and complete the square by taking 1/2 of 12 and squaring it, then add it to both sides like this - 6^2 = 36 so

(1) becomes W^2 + 12W + 36 = 225

which is (W + 6)^2 = 225

which if you take the square root of both sides is: W+6 = + 15 or - 15.

So W is 9 or -21.

Another way is to use (2) and plug it into the quadratic formula : for ax^2 + bx + c = 0,

x = -b +-Sqrt(b^2 - 4ac) / 2a

this also gives w = 9 or -21

The third way is to notice that 9 * 21 is 189 and use the usual factoring method. But that is not likely for most students since they don't learn the 21 times table. You could write down the ways to add to 12, but it seems tedious.

Either way, you get w is 9 or -21. Since length must be positive,

W = 9.

Then you can say L = W + 12 from the question itself, and

L = 21

Check it with W*L = 9*21 = 189 = the given area and you are done.

Answer 2

Length = 21, width = 9

L=12+W

LW = 189

12W+W^2 = 189

W^2 + 12W + 36 = 189 + 36

(W+6)^2 = 15^2

W = -6 + or - 15 = -21 or 9

W =9

L=12+9 = 21

Length = 21 feet, width = 9 feet