Sure! Let's solve this problem step-by-step.
We are given that the perimeter of the rectangular pen is 100 meters. The perimeter [tex]\(P\)[/tex] of a rectangle is given by the formula:
[tex]\[ P = 2 \cdot (\text{length} + \text{width}) \][/tex]
For this problem:
[tex]\[ P = 100 \][/tex]
Let's denote the width of the rectangle as [tex]\( w \)[/tex] and the length as [tex]\( l \)[/tex].
Using the perimeter formula, we can write:
[tex]\[ 2 \cdot (l + w) = 100 \][/tex]
Dividing both sides by 2, we get:
[tex]\[ l + w = 50 \][/tex]
[tex]\[ l = 50 - w \][/tex]
Hence, the length [tex]\(l\)[/tex] can be expressed in terms of the width [tex]\(w\)[/tex].
Next, we want to find the area [tex]\(A\)[/tex] of the rectangle, which is given by the product of its length and width:
[tex]\[ A = l \cdot w \][/tex]
Substituting [tex]\(l\)[/tex] from the equation [tex]\( l = 50 - w \)[/tex] into the area formula:
[tex]\[ A = (50 - w) \cdot w \][/tex]
Simplifying this, we get:
[tex]\[ A = 50w - w^2 \][/tex]
Rearranging it in standard quadratic form:
[tex]\[ A(w) = -w^2 + 50w \][/tex]
Thus, the quadratic equation that gives the area [tex]\(A\)[/tex] of the pen in terms of its width [tex]\(w\)[/tex] is:
[tex]\[ A(w) = -w^2 + 50w \][/tex]
So, the correct option is:
[tex]\[ A(w) = 50w - w^2 \][/tex]
This matches the third option in the given choices:
[tex]\[ \boxed{A(w) = 50 w - w^2} \][/tex]
