Basic Calculus Performance Task 2
SOLVING OPTIMIZATION PROBLEMS USING CALCULUS
2. A rectangular chicken pen is bounded on one side by the wall of the
house and the other three sides by 160 meters of fences. Find the
dimensions of the chicken pen if the area is a maximum.
a. What is the objective? Let it be ๐ด(๐ฅ).
The objective is the area of rectangular chicken. We are required to find the
dimensions that give the biggest area.
b. Sketch the chicken pen. Label the important parts.
c. What variable are you going to control? Let it be ๐ฅ.
As the length of the wall is fixed, let the fixed value be y and the variable we
are going to control be x. Let x be the width of the chicken pen.
d. What function accurately models this problem?
Given that the total length of the fence is 160m and with the aid of the
illustration,
We have,
2x+y=160
โ y=160โ2x
Now, the question is to find the dimensions of the chicken pen if the area is
maximum. Therefore, the area of the rectangle pen
Area,
A=xy
would be
A=x
(
160โ2x
)
=160 xโ2x2
. Let it be continuous over [0,
80].
Our model would be
A(x)=160 xโ2x2
e. What are the dimensions of the chicken pen?
Before finding the critical points, we must first find the derivative of the
function and equate it to zero and find the value of x.
A
(
x
)
=160 xโ2x2
A ' (x)=
(
1
)
160 x1โ1โ
(
2
)
2x2โ1=160โ4x
160โ4x=0โ x=40
Evaluating ๐ด at 0, 40, and 80.
x
0 40 80
y Fence
x Fence
Fence x
Wall of the
house
Chicken
pen
