A rancher plans to make four identical and adjacent rectangular pens against a barn, each with an area of 100m^2. what are the dimensions of each pen that minimize the amount of fence that must be used?
To find the dimensions of each rectangular pen that minimize the amount of fence used, we can start by understanding the requirements and constraints of the problem.
Let's assume the width of each pen is "w" meters, and the length of each pen is "l" meters. Since all four pens are identical and adjacent, the total length of the barn covered by the pens will be 4w meters.
To minimize the amount of fence used, we need to minimize the total perimeter of the pens. The total perimeter is the sum of all four sides of each pen.
Considering the area of each pen is given as 100m^2, we can use that information to set up an equation.
The equation for the area of a rectangle is: length x width = area. Therefore, l x w = 100.
We can express the total perimeter as: 2w + 2l + 4w = total perimeter.
To minimize the perimeter, we need to minimize the total length of the fence, which is given by 2w + 2l.
Now, we can express the total length of the fence, 2w + 2l, in terms of a single variable, either w or l, using the information we have.
Let's solve the area equation for l: l = 100/w.
Substituting this into the equation for the total perimeter, we have: 2w + 2(100/w).
Now, we can differentiate this expression with respect to w to find the critical point where the perimeter is minimized. The critical point can be found by setting the derivative equal to zero and solving for w.
Differentiating, we have: 2 - 200/w^2 = 0.
Simplifying, we get: 2w^2 - 200 = 0.
Solving for w, we find w = sqrt(100) = 10 meters.
Substituting w = 10 into the area equation, we get l = 100/10 = 10 meters.
Therefore, the dimensions of each pen that minimize the amount of fence used are 10 meters by 10 meters.