Max area - 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 maximum area if the total perimeter is 16 feet.
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A similar problem (with different perimeter) was solved here:
http://answers.yahoo.com/question/index?qid=20100312091756AAxFWwi
The method is the same.
To find the dimensions of a Norman window of maximum area, we can use calculus to find the maximum of the area function.
Let's start by assigning variables to the dimensions of the window:
- Width of the rectangular part of the window: x
- Height of the rectangular part of the window: y
- Radius of the semicircle: r
We know that the total perimeter is 16 feet, so we can express this information as an equation:
Perimeter = 2y + 2x + πr + 2r = 16
Since the semicircle is attached to the top of the rectangular part, the width of the rectangular part is equal to the diameter of the semicircle (2r). So, we have:
x = 2r
Substituting this into the perimeter equation, we get:
2y + 6r + πr = 16
Now, let's express the area of the window as a function of x and y:
Area = x * y + (πr^2) / 2
Since we know that x = 2r, we can rewrite the area equation as:
Area = 2r * y + (πr^2) / 2
To find the maximum of the area function, we need to differentiate it with respect to r and set the derivative equal to zero:
d(Area)/dr = 2y + πr = 0
Solving this equation gives us the value of r:
r = -2y/π
Now, let's substitute this value of r back into the perimeter equation to find y:
2y + 6(-2y/π) + π(-2y/π) = 16
2y - 12y/π - 2y = 16
-12y/π = 16
y = -16π/12
y = -4π/3
Since we cannot have negative dimensions, we reject the negative value of y. So, the dimensions of the Norman window of maximum area are:
Width (x) = 2r = 2(-4π/3) = -8π/3
Height (y) = -4π/3
However, the dimensions of a window cannot be negative, so it seems that it is not possible to construct a Norman window of maximum area given these constraints. Double-check the given information and constraints to ensure accuracy.
To find the dimensions of a Norman window of maximum area, we need to optimize the area with respect to the given constraint (perimeter = 16 feet).
Let's break down the problem into smaller steps:
Step 1: Define the variables
Let x be the width of the rectangular window, and y be the height of the rectangular window.
Step 2: Write the equations based on the given information
Since the total perimeter is equal to 16 feet, we can write an equation based on the dimensions of the window:
Perimeter = 2 * (width + height + semicircle circumference) = 16
Perimeter of the rectangular window = 2 * (x + y)
Circumference of the semicircle = π * (x/2)
So, the equation becomes:
2 * (x + y + π * (x/2)) = 16
Simplifying this equation further, we get:
x + y + π * x = 8
Step 3: Express one variable in terms of the other
Rearrange the equation to express one variable in terms of the other. Let's solve for y in terms of x:
y = 8 - x - π * x
Step 4: Calculate the area
The area of the Norman window is the sum of the area of the rectangular portion and the area of the semicircle.
Area of the rectangular portion = x * y
Area of the semicircle = (π * (x/2)^2)/2
Total area = x * y + (π * (x/2)^2)/2
Step 5: Maximize the area
To maximize the area, we need to find the critical points by taking the derivative of the area function with respect to x and setting it to zero.
Find the first derivative: d(Area)/dx = (y - (π * x)/2) + (2xy + π * (x^2)/4)/2
Set the first derivative equal to zero and solve for x:
(y - (π * x)/2) + (2xy + π * (x^2)/4)/2 = 0
Step 6: Solve for x and y
Use the equation from step 3 (y = 8 - x - π * x) and the equation from step 5 to solve for x. Then, substitute the obtained value of x into the equation from step 3 to solve for y.
Step 7: Calculate the maximum area
Substitute the values of x and y into the area formula to find the maximum area of the Norman window.
And that's how you can find the dimensions of a Norman window of maximum area by using calculus and optimization techniques.