Question Regarding System of Equations

[trulyfalse]

Hey Pf, I'm working on precalculus review and I have found myself stumped by a question.

Homework Statement

A rectangular field is enclosed by 600m of fencing. A second rectangular field, which is alongside a river, has the same area and is also enclosed by 600m of fencing. However, this second field has fencing on only three sides because there is no need for fencing along the riverbank. Create a system of quadratic equations to model the problem. (Answer: A = -x²+300x, A = -2x²+60x)

Homework Equations

ax²+bx+c

The Attempt at a Solution

I began by modeling the first field:
P = 2x+2y
600 = 2x+2y
300 = x+y
y = -x+300

A = xy
A = x(-x+300)
A = -x²+300x

Afterwards I modeled the second field:
600 = 2x+y
y = -2x+600

A = xy
A = x(-2x+600)
A = -2x²+600x

As you can see, the equations are different. I cannot see where I went wrong. Could someone please correct my folly? :)

 

[Bread18]

I can't see where you wrong, are you sure you read the answer right?

 

[trulyfalse]

Positive. It may be of use to note when I entered the equations I created into my calculator it yielded solutions of (0,0) and (0,300). Seeing as those solutions are illogical, that means it's either a bad question, or I made an error somewhere along the way.

EDIT: I just used the same method of checking for the "correct" solution; it yielded values of (-240,-129600) and (0,0). Perhaps it's just a bad question...

 

[Bread18]

Well the area of the second one should be $\geq$ the first one. And if the answers are right, that doesn't always hold true.

 

[trulyfalse]

I'll skip it and move on I suppose. :)

 

[Bread18]

Yeah that might be a good idea.

 

[Mark44]

Hey Pf, I'm working on precalculus review and I have found myself stumped by a question.

Homework Statement

A rectangular field is enclosed by 600m of fencing. A second rectangular field, which is alongside a river, has the same area and is also enclosed by 600m of fencing. However, this second field has fencing on only three sides because there is no need for fencing along the riverbank. Create a system of quadratic equations to model the problem. (Answer: A = -x²+300x, A = -2x²+60x)

Homework Equations

ax²+bx+c

The Attempt at a Solution

I began by modeling the first field:
P = 2x+2y
600 = 2x+2y
300 = x+y
y = -x+300

A = xy
A = x(-x+300)
A = -x²+300x

Afterwards I modeled the second field:
600 = 2x+y
y = -2x+600

A = xy
A = x(-2x+600)
A = -2x²+600x

As you can see, the equations are different. I cannot see where I went wrong. Could someone please correct my folly? :)

You can't use the same variables for the two fields.
Let x₁ and y₁ be the width and length of the first field (the one with fences along all four sides).
Let x₂ and y₂ be the width and length of the second field (the one with the river as one boundary).

For the first field, A = x₁*y₁ = x₁(300 - x₁)
For the second field, A = x₂*y₂ = x₂(600 - 2x₂)