To solve this problem, let’s start by understanding the constraints and how we can derive the quadratic equation for the area of the rectangular pen in terms of its width.
Step 1: Define the Perimeter Relationship
A farmer has 100 meters of fencing to enclose a rectangular pen. The perimeter (P) of the rectangular pen can be expressed as:
[tex]\[ P = 2L + 2W \][/tex]
where [tex]\( L \)[/tex] is the length and [tex]\( W \)[/tex] is the width of the rectangle. Given that the perimeter is 100 meters:
[tex]\[ 100 = 2L + 2W \][/tex]
Step 2: Express the Length in Terms of Width
We can solve the above equation for the length [tex]\( L \)[/tex]:
[tex]\[ 100 = 2L + 2W \][/tex]
[tex]\[ 2L = 100 - 2W \][/tex]
[tex]\[ L = \frac{100 - 2W}{2} \][/tex]
[tex]\[ L = 50 - W \][/tex]
Step 3: Write the Area as a Function of Width
The area [tex]\( A \)[/tex] of a rectangle is given by the product of its length and width:
[tex]\[ A = L \times W \][/tex]
Substitute the expression for [tex]\( L \)[/tex] from step 2:
[tex]\[ A = (50 - W) \times W \][/tex]
Step 4: Expand and Simplify
Now, expand the expression:
[tex]\[ A = 50W - W^2 \][/tex]
This simplifies to:
[tex]\[ A = -W^2 + 50W \][/tex]
Which matches one of the given forms:
[tex]\[ A(w) = w(50 - w) = 50w - w^2 \][/tex]
Therefore, the quadratic equation that gives the area [tex]\( A \)[/tex] of the pen as a function of its width [tex]\( w \)[/tex] is:
[tex]\[ A(w) = 50w - w^2 \][/tex]
Hence, the correct answer is:
[tex]\[ \boxed{A(w) = 50w - w^2} \][/tex]
