rancher plans to fence rect. pasture adjacent to river....

[rikaminami]

This is a bonus question and I don't really understand it. If you could help in any way, that would be greatly appreciated!

A rancher plans to fence a rectangular pasture adjacent to a river. The rancher has 100 meters of fence, and no fencing is needed along the river.
(a) Express the area A of the pasture as a function of x, the length of the side parallel to the river. What is the domain of A?
(b) Graph the area function A(x) and estimate the dimensions that yield the maxium amount of area for the pastures.
(c) Find the dimensions that yield the maximum amout of area for the pastures by completing the square.

I figured out number a. how exactly are you supposed to find the dimensions if you complete the square? help! please and thank you!

p.s. I figured out b. if i'm right :lol:

 

[stapel]

First, draw a picture, and label the indicated side with the provided variable.

a) What is the formula for the perimeter P of a rectangle with width w and length L? In your case, one of the sides, being unfenced, does not "count" in the perimeter (the length of fencing), so how should this formula be adjusted?

Now plug in the variable for the length and the value for the perimeter. Solve this equation for the width w.

What is the formula for the area A of a rectangle with width w and length L (or, in your case, x)?

Using the equation for "w=" (above), plug into the area formula to create a function A(x).

For the domain: Can area be negative?

b) This is just graphing. Find the (approximately) highest point on the picture.

c) This is just plugging into an algorithm. Follow the steps they gave you in class, and find the vertex.

If you get stuck, please reply showing all of your work and reasoning. Thank you!

Eliz.

 

[TchrWill]

Re: rancher plans to fence rect. pasture adjacent to river..

This is a bonus question and I don't really understand it. If you could help in any way, that would be greatly appreciated!

A rancher plans to fence a rectangular pasture adjacent to a river. The rancher has 100 meters of fence, and no fencing is needed along the river.
(a) Express the area A of the pasture as a function of x, the length of the side parallel to the river. What is the domain of A?
(b) Graph the area function A(x) and estimate the dimensions that yield the maxium amount of area for the pastures.
(c) Find the dimensions that yield the maximum amout of area for the pastures by completing the square.

so i think that the equation has to be a quadratic equation but I don't know how to get to it and i'm pretty sure the domain is [0,100] but I could be wrong.

A popular problem in recreational mathematics seeks the dimensions of a rectangular garden that maximizes the garden area for a given perimeter of fencing. It is very often posed in two different versions. One asks for the dimensions of a rectangular garden with maximum enclosed area for a fixed perimeter while the other asks for the same thing while using the wall of a barn as one side of the garden. Not too surprisingly, the answers are different.
Lets first address the garden with fencing on all four sides.
What are the dimensions of a rectangular garden with a fixed perimeter or 60 ft. that maximizes the enclosed area?

Considering all rectangles with the same perimeter, the square encloses the greatest area.
Proof: Consider a square of dimensions x by x, the area of which is x^2. Adjusting the dimensions by adding
"a" to one side and subtracting "a" from the other side results in an area of (x + a)(x - a) = x^2 - a^2. Thus, however small the dimension "a" is, the area of the modified rectangle is always less than the square of area x^2.
Therefore, the dimensions of the garden with maximum area for the given perimeter, p, of fencing is p/4 by p/4.
Given a length of fencing of 60 feet, the garden dimensions become 15 by 15 ft. for an area of 225 sq.ft.

Lets now assume that we have a linear barn wall to act as one edge of the rectangle we wish to enclose.
Letting the length of the side perpendicular to the barn wall equal "x", we can express the dimensions of the garden area by A = x(p - 2x) = px - 2x^2. Taking the first derivitive of A and setting it equal to zero, we derive p - 4x = 0 or 4x = p making x = p/4. While this appears to be the same answer as for the 4 sided garden, with x = p/4, the other dimension becomes p - 2(p/4) = p - p/2 = p/2. While the area of the garden fenced on all four sides was A = p^2/16, the area of the garden with fencing on only three sides becomes (p/4)(p/2) = p^2/8, literally twice the area of the square garden fenced on all four sides. Thus, for our simple example presented earlier, with a fence length of 60 ft., the maximum enclosed area using the barn wall as one side becomes 60^2/8 = 450 sq.ft.