A rancher needs to build four identical pens for his animals as shown in the diagram below. He has 240 feet of fencing. He does not need to put the fence along the barn. Find the length and width that would maximize the area of the pens. What is the maximum area for each pen?
60 feet by 40 feet
To find the dimensions that maximize the area of the pens, we can use the concept of optimization. Let's break down the problem step by step:
1. Assign variables: Let's assume the length of each pen is 'L' and the width is 'W.'
2. Determine the constraints: The rancher has a total of 240 feet of fencing available. Since there are four identical pens and we don't need to consider the fence along the barn, each pen would require (2L + W) feet of fencing.
Total fencing used = 4 × (2L + W) = 8L + 4W
We have the constraint: 8L + 4W = 240
3. Express the objective function: We want to maximize the area of each pen. The area of a rectangle is given by A = Length × Width. So, the objective function is A = L × W.
4. Solve the constraint equation for one of the variables: We can rearrange the constraint equation to obtain W in terms of L.
8L + 4W = 240 ⟹ 4W = 240 - 8L ⟹ W = (240 - 8L)/4
5. Substitute the expression obtained in step 4 into the objective function: Substitute W = (240 - 8L)/4 in the objective function A = L × W.
A = L × [(240 - 8L)/4] ⟹ A = L(240 - 8L)/4
6. Simplify the objective function: Multiply L into the bracket and simplify:
A = (240L - 8L^2)/4 ⟹ A = 60L - 2L^2
7. Differentiate the objective function: To find the maximum value, differentiate the objective function A with respect to L.
dA/dL = 60 - 4L
8. Set the derivative equal to zero and solve for L:
60 - 4L = 0 ⟹ L = 60/4 ⟹ L = 15
9. Substitute the value of L into the constraint equation to find W:
8L + 4W = 240 ⟹ 8(15) + 4W = 240 ⟹ 120 + 4W = 240 ⟹ 4W = 240 - 120 ⟹ 4W = 120 ⟹ W = 120/4 ⟹ W = 30
10. Calculate the maximum area: Substitute the values of L and W into the objective function A = L × W.
A = 15 × 30 ⟹ A = 450 square feet
Therefore, the maximum area of each pen is 450 square feet when the length is 15 feet and the width is 30 feet.