Question 918457: Dear Sir/Madam.
Can you please help me solve the following multi-part problem? I tried answering some below. Thank you.
Farmer wants to build a corral for his cows, sheeps goats and pis as shown below. He has a total of 400 feet of fencing. Please find the corral dimensions that maximize the area of the corral.
(Diagram: 1 large rectangle, divided into 4 smaller rectangles.
Length of the corral = y feet, and the width of each corral=X feet.)
____________________________
I I I I I
I I I I I Y feet long
I I I I I
I I I I I
____________________________
x feet x feet x feet x feet
1. Find an equation for the total area of all four corrals in terms of X and Y
4 XY feet^2
2. The width = X feet, length = Y feet and there are 400 feet of fencing. Find an expression for the length (Y) of the corral in terms of X.
8X + 5Y = 400
5Y = 400 - 8X
Y = 80 - 8/5X
= -8/5X + 80
3. Use answer from (2) to write an equation for the total are of all THREE corrals in terms of X. Leave answer in factored form!
Area of 3 Corrals = 3XY feet ^2
= 3X (-8/5X + 80)
= -24/5X^2 + 80/3X
Here's where I am lost!
The next 3 parts will lead to solutions for the dimensions necessary to maximize the are of the corrals using three different methods.
Method# 1 - Using X intercept/factored form of the quadratic
a. Use the X-intercept to find the axis of symmetry
b. Find the Y-coordinate of the vetex using the X-value for the axis of the symmetry
c. Sketch the graph using X-intercept/factored form
d. Using equations and graph, determine the dimensions X and Y that will yield the maximum area of the corrals.
e. Describe how to find the maximum are with your equation in x-intercept form (and find it)
Metho# 2 - Using the vertex form of a quadratic
a. Multiply your equation for are into general form and then convert in into vertex form by completing the square.
b. Sketch the graph. You may use the x-intercepts from Method # 1
c. Use your equation and /graph to determine the value of X and y that will yeild the maximum area. Describe how to use your equation to find these values.
d. Describe how to find the maximum area with your equation in vertex form.
Method # 3 - General Form
a. Write the equation for are in terms of x in general form
b. Find the x-coordinate of the vertex using the formula for the axis of symmetry (x = -b/2a)
c. Use your equation/graph to determine the value of x and y that will yield the maximum area. Describe how to use your equation to find these values.
d. Describe how to find the max. area with your equation in general form.
Lastly, select most favourite method of solving and explain why. Compare and contrast the methods . Talk about advantages and disadvantages of each method.
HELP!!!!!! This should not be so difficult. Maybe I am not understanding the question.
THANK YOU!!!!
Answer by josgarithmetic(39617)   (Show Source):
You can put this solution on YOUR website! Let x and y be the overall outside dimensions of the rectangle fields combined. One would imagine that the fencing used will make an intersection in the middle of the rectangular region and the parts of the fencing that connect to the outer sides meet them at right angles.
The length of fencing can be represented as  ;
 when simplified.
Your goal for the question is to find the maximum area, of the entire fenced region.
Let A be the total area. You still just use the dimensions of the whole area, x and y;
 .
(1)
Solve the fence equation for either variable in terms of the other.
(2)
Substitute for y into the area A equation.
This area formula is quadratic, and has roots. Note that if you perform the multiplication on the right-side of the equation, the coefficient on  term will be  , indicating that A has a maximum for its vertex. This vertex, the maximum A, occurs in the exact middle of the roots of A. The roots of A will be the values of x for which A=0.
(3)
Find the zeros for maximum A.
Either  OR  .
For the binomial part,
-
The maximum area will be at exact middle of x at 0 and x at 400/3;
which will be for  .
-
The corresponding y will be,
Both x and y, each are when reduced,  , which forms a square shape.
The corral dimensions:  length by width.
This is a square of side length  feet.
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