Critical points and of polynomial functions

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Homework Help Overview

The problem involves determining the dimensions of a rectangular region with a fixed area of 125,000 square feet, where different costs are associated with the fencing used on different sides. The objective is to minimize the total cost of the fencing.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants discuss the relationship between area, perimeter, and cost, questioning how to derive the perimeter from the area and the cost per foot of fencing. There are attempts to clarify the relevance of the cost variables in relation to the dimensions of the rectangle.

Discussion Status

Some participants have begun to express equations for perimeter and cost based on the dimensions, while others are still grappling with how to connect the area to these equations. There is an ongoing exploration of how to express one variable in terms of another to simplify the problem.

Contextual Notes

Participants note confusion regarding the distinction between cost per foot and cost per square foot, which may affect their understanding of the problem setup. There is also mention of needing to focus on the cost of fencing rather than the perimeter itself.

cptstubing
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Homework Statement


A rectangular region of 125,000 sq ft is fenced off. A type of fencing costing $20 per foot was used along the back and front of the region. A fence costing $10 per foot was used for the other sides. What were the dimensions of the region that minimized the cost of the fence?

Homework Equations


A=l*w

The Attempt at a Solution


I haven't a clue where to begin with this. What should I be thinking?
 
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cptstubing said:

Homework Statement


A rectangular region of 125,000 sq ft is fenced off. A type of fencing costing $20 per foot was used along the back and front of the region. A fence costing $10 per foot was used for the other sides. What were the dimensions of the region that minimized the cost of the fence?

Homework Equations


A=l*w

The Attempt at a Solution


I haven't a clue where to begin with this. What should I be thinking?
Perimeter.

Think Perimeter.
 
SammyS said:
Perimeter.

Think Perimeter.

I know this question should eventually look like a regular function like the practice questions I've been doing for ages, but I'm failing to see how I get there.
 
cptstubing said:
I know this question should eventually look like a regular function like the practice questions I've been doing for ages, but I'm failing to see how I get there.
What's the perimeter of a rectangle?

You have two dimensions to work with, length and width.
 
SteamKing said:
What's the perimeter of a rectangle?

You have two dimensions to work with, length and width.

L*W = area in sq feet, which is 125000.
But the variables, $20 and $10 per square foot of fence, don't seem relevant.
I don't see how I can get a perimeter from area=125000 and the price per foot of fence.
I do appreciate help with this.
 
cptstubing said:
L*W = area in sq feet, which is 125000.
But the variables, $20 and $10 per square foot of fence, don't seem relevant.
I don't see how I can get a perimeter from area=125000 and the price per foot of fence.
I do appreciate help with this.

For right now, forget about the area of the rectangle. That comes later.

If you have a rectangle, any rectangle, with a length L and a width W, what is the formula for the perimeter of this shape?
 
SteamKing said:
For right now, forget about the area of the rectangle. That comes later.

If you have a rectangle, any rectangle, with a length L and a width W, what is the formula for the perimeter of this shape?

Perimeter would be (L*2)+(W*2)
 
cptstubing said:
L*W = area in sq feet, which is 125000.
But the variables, $20 and $10 per square foot of fence, don't seem relevant.
I don't see how I can get a perimeter from area=125000 and the price per foot of fence.
I do appreciate help with this.

When you write L*W, what do you mean? What is L? What is W? So, if you did know L and W, what other aspects of the fencing could you compute using those values?
 
Ray Vickson said:
When you write L*W, what do you mean? What is L? What is W? So, if you did know L and W, what other aspects of the fencing could you compute using those values?

L*W means Area, or 125000. Length * Width. If I knew the two variables, I'd know the perimeter.
 
  • #10
cptstubing said:
L*W means Area, or 125000. Length * Width. If I knew the two variables, I'd know the perimeter.
You know the two variables -- L and W. You just don't happen to know their values. Even so, you should be able to write one expression that represents the perimeter of the fence, and another that represents the cost of that fence.
 
  • #11
Mark44 said:
You know the two variables -- L and W. You just don't happen to know their values. Even so, you should be able to write one expression that represents the perimeter of the fence, and another that represents the cost of that fence.

P=(L*2)+(W*2)
Cost =(L*2*10)+(W*2*20)
I don't know if this is correct. What is I need to know before I can do this? A process or something?
 
  • #12
cptstubing said:
P=(L*2)+(W*2)
Cost =(L*2*10)+(W*2*20)
I don't know if this is correct.
That's a good start.

cptstubing said:
What is I need to know before I can do this? A process or something?
Forget process -- what you need to do is think about this problem.

You have another equation that involves the known area. Use it to solve for one of the variables in terms of the other. Then you can replace a variable in your cost equation, turning it into a function of a single variable.
 
  • #13
cptstubing said:
L*W = area in sq feet, which is 125000.
But the variables, $20 and $10 per square foot of fence, don't seem relevant.
I don't see how I can get a perimeter from area=125000 and the price per foot of fence.
I do appreciate help with this.
That's $20 and $10 per foot. It's not per square foot .
 
  • #14
cptstubing said:
P=(L*2)+(W*2)
Cost =(L*2*10)+(W*2*20)
I don't know if this is correct. What is I need to know before I can do this? A process or something?

The farmer does not care about the perimeter; he just cares about the cost of fencing and the area enclosed. You know (that is, have expressions for) both of these in terms of L and W, so you are close to done. Can you see what to do next?
 

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