zjb0417 said:
The length of a rectangular field is 2 yards more than its width. Find the width if the area of the field is 120 yd2.
1. Explain why the answer of a width of –12 yards is unreasonable.
I want to first reply to your question regarding a -12 as width. The reason why -12 CANNOT be the width is because width is a length and length indicates distance and distance CANNOT be negative. For example, you don't say: The cat is negative 20 feet away from where I am standing, right? You would say: The cat is 20 feet away from where I am standing.
Back to your question.
We have this:
The length of a rectangular field is 2 yards more than its width. Find the width if the area of the field is 120 yd^2.
The question tells you that the length is 2 more yards than the width but it does not give you the width, right?
So, let x = the length of the width (in yards) and since the length is 2 more than the width, we can write the length as x + 2 yards.
They give you the area: 120 yards^2, which is read "120 square yards." Why do we need a square there? Area is measured in square units.
The area of a rectangle formula is: Area = length times width.
We have everything we need to
PLUG AND CHUG.
Area = 120 yd^2
width = x
length = x + 2
120 = (x) (x + 2)
Simplify and solve for x.
120 = x^2 + 2x
Subtract 120 from both sides of the equation and set the equation to = 0.
x^2 + 2x - 120 = 0
We now have a quadratic equation. Factor the quadratic equation. Do you see -120? Ask yourself: What two numbers can be multiplied to give -120 but when added will produce the middle coefficient 2? Do you see 2? How about 12 times -10?
Well, 12 times -10 = -120 but at the same time, 12 + (-10) = 2.
So, x^2 + 2x - 120 = 0 becomes: (x + 12) (x - 10) = 0
We now have two factors: (x + 12) and (x - 10).
Set each factor to = 0 and solve for x INDIVIDUALLY.
x + 12 = 0
x = -12....We reject this answer because distance
MUST BE POSITIVE.
Set next factor to equal 0:
x - 10 = 0
x = 10
Like I said above, x represents our width.
Final answer: width = 10 yards
