I really hate to dump a problem like this on the teachers here, but I really need to get an answer to this. Thank you and I'm sorry.
A Norman window is constructed by adjoining a semicircle to the top of an ordinary rectangular window. Find the dimensions of a Norman window of a maximum area if the total perimeter is 52 feet.
Oh, i apologize if this gives the impression I want you to solve it for me. I would just like to know where to start this from, then I can take it from there.
as an answer i used x as the width of the rectange (and diameter of the half circle)and used h as the height of the rectangle. i got the answer of x=.409155 and h= 25.4741. how did i do?
I would use 2x as the with and have the radius of the circle be x.
Your h would equal (1/2)(52-2x-3.14x)
I think the x and h should be closer
The closer the numbers, the max the area
No problem at all! I'm here to help you. To find the dimensions of a Norman window with maximum area, given a total perimeter of 52 feet, we can use calculus to find the solution. Let's break it down step by step:
Step 1: Understand the problem
In a Norman window, a semicircle is joined to the top of a rectangular window. We need to find the dimensions (length and width) of the rectangular part that maximize the total area while having a perimeter of 52 feet.
Step 2: Define the variables
Let's assume the length of the rectangle is 'L' and the width is 'W'.
Step 3: Formulate the equations
We know that the perimeter of a rectangle is given by:
Perimeter = 2L + 2W
In our case, since the semicircle is added to the top of the rectangle, the combined perimeter should be 52 feet. So we can write:
2L + 2W + πL = 52
The area of the Norman window is given by the sum of the rectangle's area and half the area of the semicircle:
Area = LW + (πL^2)/2
Step 4: Simplify the equation
We can simplify the perimeter equation by factoring out 'L' and then rearranging:
2L + 2W + πL = 52
(2 + π)L + 2W = 52
L = (52 - 2W) / (2 + π)
Step 5: Substitute the simplified equation into the area equation
Substitute the expression we obtained for 'L' into the area equation:
Area = (52 - 2W) / (2 + π) * W + (π[(52 - 2W) / (2 + π)]^2)/2
Step 6: Take the derivative of the area equation
To find the maximum area, we need to find where the derivative of the area equation is equal to zero. Differentiating the equation with respect to 'W' will give us that critical point:
d(Area)/dW = 0
Step 7: Solve for 'W'
Solve the derived equation from step 6 for 'W' to find the value that will maximize the area.
Step 8: Substitute 'W' back into the perimeter equation
Use the value of 'W' obtained in step 7 and substitute it into the perimeter equation:
2L + 2W + πL = 52
Step 9: Solve for 'L'
Solve the equation from step 8 for 'L' to find the corresponding length.
Now you have the dimensions of the Norman window with maximum area. Remember, the maximum area occurs when the derivative is zero, so make sure to double-check that it is a maximum and not a minimum.