This is my problem:
A carpenter is building a rectangular room with a fixed perimeter of 336 ft. What dimensions would yield the maximum area? What is the maximum area?
I am confused on what steps it takes to come up with the answer. I would like an example if possible with each step included.
P = 2L + 2W
Find possible length and width of this rectangle.
Hint: Start with a square with 84 feet on each side. That gives an area of 7,056 square feet.
Try other dimensions.
Yes, but what are the steps to find the answer? I do not have an example in my text.
Follow the directions I posted.
A = L * W
To find the dimensions that yield the maximum area for a rectangular room with a fixed perimeter of 336 ft, we can use the concept of optimization. Let's break down the problem into steps to better understand the process:
Step 1: Identify the variables:
To solve this problem, we need to define our variables. In this case, the variables are the length and width of the rectangular room. Let's say the length of the room is L and the width is W.
Step 2: Formulate the constraints:
The problem states that the perimeter of the room should be fixed at 336 ft. The perimeter of a rectangle is calculated by adding up all four sides, which is given by:
2L + 2W = 336
Step 3: Express the objective function:
We want to find the maximum area of the rectangular room. The area of a rectangle is given by multiplying the length and width, which is:
Area = L * W
Step 4: Solve for one variable in terms of another:
Since our constraint equation has two variables (L and W), we need to solve one of them in terms of the other, so we can express the objective function in terms of a single variable.
From our constraint equation, we can solve for L in terms of W:
2L + 2W = 336
2L = 336 - 2W
L = (336 - 2W) / 2
L = 168 - W
Step 5: Substitute the expression from step 4 into the objective function:
Now, substitute the expression for L from step 4 into the objective function from step 3:
Area = L * W
Area = (168 - W) * W
Area = 168W - W^2
Step 6: Maximize the objective function:
To find the maximum area, we need to maximize the objective function. This can be done by finding the critical points of the function, which occur when the derivative of the function is zero.
Taking the derivative of the objective function:
d(Area) / dW = 168 - 2W
Setting the derivative equal to zero:
168 - 2W = 0
2W = 168
W = 168 / 2
W = 84
Step 7: Find the corresponding value of the other variable:
Now that we know the width W is 84 ft, we can substitute it back into the expression we derived for L in step 4:
L = 168 - W
L = 168 - 84
L = 84
Step 8: Check for maximum using the second derivative:
To ensure that we have found the maximum value, we need to analyze the second derivative of the objective function. But in this case, we can skip this step since we only have one critical point, which means it is the only maximum.
Step 9: Calculate the maximum area:
Now, substitute the values of L and W into the objective function to find the maximum area:
Area = L * W
Area = 84 * 84
Area = 7056 ft^2
So, the dimensions that yield the maximum area for a rectangular room with a fixed perimeter of 336 ft are 84 ft by 84 ft, and the maximum area is 7056 sq ft.