Calculus question story problem?

Question

One hundred feet of fencing is used to enclose three sides of a rectangular pasture. The side of a barn closes off the fourth side. Let x be the length of one side of the fence perpendicular to the barn.

a) find and algebraic representation A(x) for the area of the pasture.

b) draw a complete graph of y= A(x)

c) What are the domain and range of y=A(x)

d) what values of x make sense in the problem situation?

Answer 1

If x is the length of one side, the fence used could be written as:

x + x + 100-2x

or

x + (100-x)/2 + (100-x)/2

for simplicity, I'll use the first one.

So, let's say the length is x, the width is (100-2x)

so the area is:

A=(x)(100-2x)

or:

A = 100x - 2x^2

That's the equation of a parabola opening downward.

Since you want the maximum, take the derivative (to find slope) and set it to zero. (where slope is zero, you have a local extrema).

dA/dx = 100 - 4x

100 - 4x = 0

4x = 100

x = 100/4 = 25

Since this is a physical situation, it only makes sense for values of X where A is non-negative.

Since A = (x)(100-2x), the zeros are where A = 0; x=0 or (100-2x)=0

So:

A = 0 at x = 0 or x = 50.  A is positive for all values between x=0 and x=50.

Going back to the beginning, your sides are x, 100-2x and x, with the third being closed off by the barn.

Your pasture is therefore 25 feet by 50 feet, or 1250 feet square.

a) Algebraic representation of A(x):

y = -2x^2 + 100x

b) Graph: that's up to you.

c) domain: { - infinity : + infinity}

range: { - infinity : +1250 }

d) X only makes sense for values 0 <= x <= 50.