One hundred feet of fencing is available to make a rectangular dog pen against an existing wall. What are the dimensions of the pen that will yield the maximum area?
I need all of the steps to get to the answer.
I told you that the maximum area is a square.
Find the length of a square with a perimeter of 100. Then square that number to find the area.
To find the dimensions of the rectangular dog pen that will yield the maximum area, we can use the concept of optimization.
Step 1: Understand the problem
We have 100 feet of fencing available and need to determine the dimensions of the pen that will result in the maximum area. The pen will have three sides (two parallel sides and one side against an existing wall), so the dimensions we need to find are the length and width of the rectangular pen.
Step 2: Identify the variables
Let's assign variables to the dimensions of the pen. We can use 'l' for the length and 'w' for the width.
Step 3: Formulate the objective function
The objective is to maximize the area of the dog pen. The area of a rectangle is given by the formula A = l * w, so our objective function is A = l * w.
Step 4: Determine the constraints
We have the constraint that the total length of the fencing available is 100 feet. This can be represented by the equation 2l + w = 100, where 2l represents the two parallel sides and w represents the side against the existing wall.
Step 5: Solve the constraint equation for one variable
Solving the constraint equation for one variable will allow us to express it in terms of a single variable. Let's solve the equation 2l + w = 100 for w.
Subtracting 2l from both sides, we have w = 100 - 2l.
Step 6: Substitute the expression for w in the objective function
Substituting the expression for w in the objective function A = l * w, we get A = l * (100 - 2l).
Step 7: Simplify the objective function
To simplify the objective function, we multiply l by 100 and -2l. This gives us A = 100l - 2l^2.
Step 8: Maximize the objective function
To maximize the area, we need to find the value of l that maximizes the objective function A = 100l - 2l^2. We can do this by taking derivative of A with respect to l, setting it equal to zero, and solving for l.
Differentiating A = 100l - 2l^2, we get dA/dl = 100 - 4l.
Setting dA/dl = 0, we have 100 - 4l = 0.
Solving this equation for l, we find l = 25.
Step 9: Calculate the corresponding value of w
Substituting the value of l back into the constraint equation, we have 2(25) + w = 100. Solving for w, we get w = 50.
Step 10: Verify the solution
To verify that this solution yields the maximum area, we can use the second derivative test. Differentiating dA/dl = 100 - 4l once more, we get d²A/dl² = -4.
Since the second derivative is negative, this confirms that the value of l = 25 yields a maximum area.
Step 11: State the answer
The dimensions of the rectangular dog pen that will yield the maximum area are a length of 25 feet and a width of 50 feet.