A Norman window has the shape of a rectangle surmounted by a semicircle. (Thus the diameter of the semicircle is equal to the width of the rectangle.) If the perimeter of the window is 24 ft, find the value of x so that the greatest possible amount of light is admitted.
Damon did same question, except the perimeter was 30
easy to change
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Let's break down the problem step-by-step:
Step 1: Understand the problem.
A Norman window is composed of a rectangle with a semicircle on top. The perimeter of the window is 24 ft.
Step 2: Assign variables.
Let x be the width of the rectangle, which is also the diameter of the semicircle.
Step 3: Calculate the perimeter.
The perimeter of the window is the sum of the four sides of the rectangle (2 lengths and 2 widths) plus half the circumference of the semicircle.
Perimeter = 2(length) + 2(width) + 1/2(circumference of semicircle)
Since the width of the rectangle and the diameter of the semicircle are the same (x), we can replace them in the equation:
24 = 2(length) + 2(x) + 1/2(πx)
Step 4: Solve for length.
To find the length, we need to isolate it on one side of the equation:
24 = 2(length) + 2x + (πx)/2
Rearranging the equation, we get:
2(length) = 24 - 2x - (πx)/2
Now, divide both sides by 2:
length = (24 - 2x - (πx)/2) / 2
Step 5: Calculate the area of the window.
The area of the window can be calculated by adding the area of the rectangle and half the area of the semicircle.
Area = Area of rectangle + 1/2(Area of semicircle)
Area of rectangle = length * width = length * x
Area of semicircle = (πr^2) / 2 = (π(x/2)^2) / 2 = (πx^2) / 8
Therefore,
Area = length * x + (πx^2) / 8
Step 6: Find the value of x for the maximum amount of light.
To find the value of x that maximizes the amount of light admitted, we need to maximize the area. This can be done by finding the derivative of the area equation with respect to x and setting it equal to zero.
d(Area)/dx = x/2 + (πx)/8 = 0
Multiply through by 8:
4x + πx = 0
Factor out x:
x(4 + π) = 0
Since the width of the window cannot be zero, we can solve for x:
x = 0 / (4 + π)
The value of x for the maximum amount of light admitted through the Norman window is x = 0 ft.
Step 7: Answer.
Based on the calculations, we found that there is no value of x that maximizes the amount of light admitted through the Norman window.
To find the value of x that maximizes the amount of light admitted through the Norman window, we need to determine the dimensions of the rectangle and semicircle.
Let's start by defining the width of the rectangle as x. Since the diameter of the semicircle is equal to the width of the rectangle, the radius of the semicircle will be x/2.
The height of the rectangle is not given in the problem, so let's call it y.
The perimeter of the window can be calculated by adding the lengths of all its sides. The perimeter is given as 24 ft, so we can set up the following equation:
Perimeter = Length of rectangle + Length of semicircle + Length of rectangle
24 = 2x + π * (x/2) + 2y
Simplifying the equation, we have:
24 = 2x + (π/2)x + 2y
Now, we want to maximize the amount of light admitted, which means we want to maximize the area of the window. The area of the window can be calculated by summing the areas of the rectangle and the semicircle.
Area = Area of rectangle + Area of semicircle
Area = x * y + (π * (x/2)^2)/2
Since we're now dealing with the area, let's denote it as A.
A = x * y + (π/8) * x^2
Now we have two equations that we can work with:
24 = 2x + (π/2)x + 2y
A = x * y + (π/8) * x^2
To maximize the area A, we can use calculus by taking the derivative of A with respect to x, setting it equal to zero, and solving for x. However, in this case, we can simplify the problem by solving for y in the first equation and substituting it into the second equation.
First, solve the first equation for y:
24 = 2x + (π/2)x + 2y
24 = (2 + π/2)x + 2y
2y = 24 - (2 + π/2)x
y = (24 - (2 + π/2)x)/2
Substitute this expression for y into the second equation:
A = x * [(24 - (2 + π/2)x)/2] + (π/8) * x^2
Simplify the equation for A:
A = (12x - (1 + π/4)x^2) + (π/8) * x^2
Now we have a simplified equation for the area A in terms of x. To find the value of x that maximizes A, we need to find the value of x where the derivative of A with respect to x is equal to zero.
Take the derivative of A with respect to x:
dA/dx = 12 - (1 + π/4) * 2x + (π/8) * 2x
Set the derivative equal to zero and solve for x:
12 - (1 + π/4) * 2x + (π/8) * 2x = 0
12 - (1 + π/4 - π/4) * 2x = 0
12 - (1 - π/4) * 2x = 0
12 - (4 - π)/2 * x = 0
Simplifying further:
12 - (4 - π)/2 * x = 0
12 = (4 - π)/2 * x
x = (12 * 2)/(4 - π)
x = 24/(4 - π)
So, the value of x that maximizes the amount of light admitted through the Norman window is x = 24/(4 - π) (approximately 7.02 ft).