Final answer:
To maximize the area of each rectangular pen, find the dimensions that use the given fencing length most efficiently. Solve the equation for one variable, substitute it into the area formula, and find the dimensions that maximize the area using calculus.
Step-by-step explanation:
To maximize the area of each rectangular pen, we need to find the dimensions that use the given fencing length most efficiently. Let the length of each pen be L and the width be W.
Since each pen has 4 sides and there are 4 pens, the total length of the sides will be 4L and the total width will be 4W. We also know that the total length of the fencing available is 500m. So, we have the equation 4L + 4W = 500.
To find the dimensions that maximize the area, we can solve this equation for one variable and substitute it into the area formula. Let's solve the equation for L:
4L = 500 - 4W
L = (500 - 4W)/4
Now, we can substitute this value for L in the area formula: A = L * W = ((500 - 4W)/4) * W.
To find the dimensions that maximize the area, we can take the derivative of the area formula with respect to W and set it equal to zero to find the critical points. Then, we can evaluate these critical points to find the dimensions that give the maximum area.