Expert: Paul Klarreich - 1/23/2008

Question
Hi again. I just had another question or two about my algebra homework. It has to do with functions again.

1. A rancher with 750 feet of fencing wants to enclose a rectangular area and then divide it into four pens with fencing parallel to one side of the rectangle.
a. find a function that models the total area of the four pens.
b.  find the largest possible total area of the four pens.

2. The sum of two positive numbers is 60. Find a function that models their product 'P' in terms of 'x', one of the numbers.

If you could help me out with these i would really appreciate it! Thanks, carly

Answer
Questioner:   Carly
Category:  Advanced Math
Private:  No
 
Subject:  functions
Question:  Hi again. I just had another question or two about my algebra homework. It

has to do with functions again.

1. A rancher with 750 feet of fencing wants to enclose a rectangular area and then

divide it into four pens with fencing parallel to one side of the rectangle.
a. find a function that models the total area of the four pens.
b.  find the largest possible total area of the four pens.

2. The sum of two positive numbers is 60. Find a function that models their product

'P' in terms of 'x', one of the numbers.

If you could help me out with these i would really appreciate it! Thanks, carly
......................................................
1. A rancher with 750 feet of fencing wants to enclose a rectangular area and then

divide it into four pens with fencing parallel to one side of the rectangle.
a. find a function that models the total area of the four pens.
b.  find the largest possible total area of the four pens.

Try this diagram:

 x      x     x     x
+-----+-----+-----+-----+
|     |     |     |     |
|     |     |     |     |
|y    |y    |y    |y    | y
|     |     |     |     |
|     |     |     |     |
+-----+-----+-----+-----+
  x    x     x     x

a. find a function that models the total area of the four pens.

>> you mean 'of one variable', right?

Now the area of each pen is xy and the total is 4xy.  
Since the total fencing is 8x + 5y = 750, we solve for y:

5y = 750 - 8x

y = 150 - 8x/5

A = 4xy = 4x(150 - 8x/5)

A(x) = 600x - 32x^2/5  << your function.

b.  find the largest possible total area of the four pens.

This takes some arithmetic.  The graph of A is a parabola, wih a = -32/5, b = 600.

The vertex of a parabola is at  

x = -b/2a = -600/(-64/5)

=  3000/32 = 375/8

Now substitute that into A and do the arithmetic.

............................................

2. The sum of two positive numbers is 60. Find a function that models their product

'P' in terms of 'x', one of the numbers.

Let  x and  60-x be the numbers.

P = x(60-x).