A norman window is constructed by adjoining a semicircle to the top of an ordinary rectangular window. Find the dimensions of a Norman window with maximum area if the total perimeter is 16 feet.
X = the width of the rectangle.
Y = the length of the rectangle.
X/2 = the radius of the circle.
Please show steps.
First let's detemine one of the dimensions, length or width in terms of the other for this perimeter.
Perimeter P=x+2y+pi*x/2 so y=16-(x++pi*x/2)/2
The area Q of the semi-circle is Q=pi*(x/2)^2
The area R of the rectangle is R=x*y=x*(16-(x++pi*x/2)/2)
The total area is A=Q+R=pi*(x/2)^2 + x*(16-(x++pi*x/2)/2)
Find dA/dx and solve for x= 0, be sure to determine the max range for x and check the endpoints too.
Now, how would you find the max range?
I didn't fully understand the lesson I had on this and even though it's starting to make sense I'm still confused.
What are the max and min values for x. Most likely the answer is not at the endpoints for this problem, but when doing optimization problems they must be checked too. Frequently the optimum is at the endpoint for the domain.
I think the endpoints for the domain here are 0 and 16, so the optimum value should be somewhere in between them.
The hardest part of this problem should be trying to state it in one variable. After you have A(x), find A'(x) and solve when it's 0.
I notice I have the area of the semicircle wrong. It should be
Q=(1/2)*pi*(x/2)^2
So what would be the revised equation?
Here's the original post you gave.
"if the total perimeter is 16 feet.
X = the width of the rectangle.
Y = the length of the rectangle.
X/2 = the radius of the circle."
Perimeter P=x+2y+pi*x/2 so y=16-(x++pi*x/2)/2
I gave the wrong formula for Q, it should be half that amount.
The area Q of the semi-circle is Q=(1/2)*pi*(x/2)^2
The area R of the rectangle is R=x*y=x*(16-(x++pi*x/2)/2)
The total area is A=Q+R=(1/2)*pi*(x/2)^2 + x*(16-(x++pi*x/2)/2)
Be sure to draw a diagram and label the parts. Show what the area and perimeter of the rectangle and semicircle should be.
I 'think' I gave the correct formula for A(x) now. Simplify the right hand side, differentiate, set it to 0 and solve for x. Then go back to the perimeter formula and determine y.
Since a square encloses the maximum area for a given perimeter, let the sides of the square portion be x and then P = 3x + xPi/2 = 16.
Solve for x.
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No, that won't help.
Let's try this again.
Since a square encloses the maximum area for a given perimeter, let the sides of the square portion be x and then P = 3x + xPi/2 = 16.
Solve for x:
3x + xPi/2 = 16
3x = 16 - xPi/2
3x = 16 - (x*Pi)/2
3x = 16 - (x*3.14)/2
3x = 16 - (1.57x)
4.57x = 16
x = 16/4.57
x = 3.49
Now that we have x, we can calculate y:
y = 16 - (3.49 + (3.49*3.14)/2)
y = 16 - (3.49 + 5.22)
y = 7.29
To find the dimensions of a Norman window with maximum area, we need to first express the area in terms of one variable.
Let's start with the given perimeter equation:
P = x + 2y + (π*x/2)
Since we are looking for the maximum area, we want to express the area in terms of x only.
The area of the semicircle is A_s = (1/2)π(x/2)^2.
The area of the rectangle is A_r = x * y.
The total area is A = A_s + A_r.
Substituting the values of A_s and A_r into the equation for A, we get:
A = (1/2)π(x/2)^2 + x(16-x-(π*x/2)/2)
Simplifying further:
A = (1/4)πx^2 + x(16 - x - (π/4)x)
Now, we need to find dA/dx, set it equal to 0, and solve for x.
dA/dx = (1/2)πx - 2x - (π/4)x + 16 - x
Setting dA/dx equal to 0:
(1/2)πx - 2x - (π/4)x + 16 - x = 0
Simplifying further:
(5/4)πx - 3x + 16 = 0
To solve this equation, we can use numerical methods or estimation techniques.
Assuming we find the value of x that satisfies this equation, we can substitute it back into the perimeter equation to find y.
P = x + 2y + (πx/2) = 16
Substituting the value of x, solve for y.
This will give you the dimensions of the Norman window with maximum area.
I apologize for any confusion caused. Let's focus on finding the maximum area of the Norman window.
To find the maximum area, we need to determine the value of x that maximizes the area A. First, let's rewrite the equations:
Perimeter P = x + 2y + πx/2 = 16
Area of the semicircle Q = (1/2)π(x/2)^2
Area of the rectangle R = xy = x(16-(x+πx/2)/2)
The total area A = Q + R = (1/2)π(x/2)^2 + x(16-(x+πx/2)/2)
We can rewrite the perimeter equation in terms of y:
y = (16 - (x + πx/2))/2
Now, let's simplify the equation for A:
A = (1/2)π(x/2)^2 + x(16 - (x + πx/2)/2)
= (1/2)π(x^2/4) + x(16 - x - πx/4)
= (1/8)πx^2 + (16x - x^2 - (π/4)x^2)
To find the maximum area, we need to find the critical points by differentiating A with respect to x and setting it to zero:
dA/dx = (1/8)π(2x) + (16 - 2x - (π/2)x) = 0
Simplifying this equation:
(2/8)πx + (16 - 2x - (π/2)x) = 0
(π/4)x + 16 - 2x - (π/2)x = 0
(π/4 - π/2 - 1)x + 16 = 0
((3π - 2π - 4)/4)x + 16 = 0
(-π/4)x + 16 = 0
x = -64/π
Since the domain of x is 0 ≤ x ≤ 16, we can eliminate the negative value.
Therefore, the dimension of the rectangle x = -64/π does not make sense in this context.
Now, we should check the endpoints of the domain:
When x = 0:
y = (16 - (0 + π(0)/2))/2 = 8
When x = 16:
y = (16 - (16 + π(16)/2))/2 = 0
We have found that the maximum area occurs at the endpoint x = 16. The corresponding y-value is y = 0.
Therefore, the dimensions of the Norman window with maximum area are X = 16, Y = 0, and X/2 = 8.