Thread: farmers field word problem

Mr. Farmer has a problem that he knows can be solved mathematically. He owns land and wants to fence in a rectangular portion for his horses. He has 200 feet of fence. He could guess at the measurements that would give his horses the largest grazing area but why should he guess when he has you to help him solve this problem with algebra.

1. Determine how he can maximize his area. Set up an equation that will maximize the area of the rectangular field

2. Graph the area function.

3. If Mr. Farmer wants to enclose the largest possible rectangular area,
a. What should the length of the field be?
b. What should the width of the field be?
c. What is the maximum area that these dimensions will yield?

4. What is the meaningful DOMAIN? Explain why you picked these values.

5. What is the meaningful RANGE? Explain why you picked these values.

Originally Posted by shantellea

Mr. Farmer has a problem that he knows can be solved mathematically. He owns land and wants to fence in a rectangular portion for his horses. He has 200 feet of fence. He could guess at the measurements that would give his horses the largest grazing area but why should he guess when he has you to help him solve this problem with algebra.

1. Determine how he can maximize his area. Set up an equation that will maximize the area of the rectangular field

2. Graph the area function.

3. If Mr. Farmer wants to enclose the largest possible rectangular area,
a. What should the length of the field be?
b. What should the width of the field be?
c. What is the maximum area that these dimensions will yield?

4. What is the meaningful DOMAIN? Explain why you picked these values.

5. What is the meaningful RANGE? Explain why you picked these values.

this is an in-depth problem. it has the feeling of being some sort of assignment. why don't you tell us what you tried, what you're having trouble with?

do you realize this is an optimization problem?

do you realize you need to set up two equations with two unknowns?

do you realize that you need derivatives?

yes i relize this is an indepth problem. i know the formulas for permenter and area. my problem is just trying to set up the equations. i am at lost even where to start. i know i need two equations but that is about it. i really dont want anyone to solve this for me but maybe to help set me in the right direction with the equation. i can figure out the rest but for some reason setting up equations are really hard for me.

Originally Posted by shantellea

yes i relize this is an indepth problem. i know the formulas for permenter and area. my problem is just trying to set up the equations. i am at lost even where to start. i know i need two equations but that is about it. i really dont want anyone to solve this for me but maybe to help set me in the right direction with the equation. i can figure out the rest but for some reason setting up equations are really hard for me.

ok, i will help you to set up. but i really cannot provide you with too much explicit help, so don't hate me.

Let us begin. We are talking about a rectangle. it seems obvious here that the important components will be the length and the width of the rectangle, so let's begin there.

Let the length of the fence be
Let the width of the fence be

We get the perimeter of the fence by . (no problems here, right?)

we are told that the farmer has 200 feet of fence, so the perimeter is 200 feet. thus our first equation is:

.................(1)

this is what we call our "constraint equation," since it limits us in some way. we use this equation to solve for one variable in terms of the other, for use in substituting in another simultaneous equation, which we call the "objective equation." the objective equation is an equation that relates our objective to our unknowns. here our objective deals with area. the objective equation, then, should also be obvious. let be the area of the rectangle. then our second equation is:

.........................(2)


So there's your set up. Now what?

at this point i know there are two equations with two unknowns. i also know that a square (i've been told its a type of rectangle) will give the biggest area. at this time i know that 2(x+y)= 200 which is the peremeter (and will have 50 feet of fence on each side) and that x*y is the area. is there a way to combine these formulas that i already have into a equation? Am i on the right path?