A Norman window is constructed by adjoining a semicircle to the top of a rectangular window as shown in the figure below. If the perimeter of the Norman window is 24 ft, find the dimensions that will allow the window to admit the most light.
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To find the dimensions that will allow the window to admit the most light, we need to optimize the area of the window.
Let's denote the width of the rectangular part of the window as x, and the radius of the semicircle as r.
The perimeter of the Norman window is 24 ft, so we can write the equation for the perimeter as:
Perimeter = 2 * width of rectangle + circumference of semicircle
24 = 2x + πr + 2r
To optimize the area, we need to express the area in terms of a single variable. The area of the window is the sum of the area of the rectangle and the area of the semicircle.
Area = area of rectangle + area of semicircle
Area = x * height + (1/2) * π * r^2
We need to express the height of the rectangle in terms of x and r. Since the top of the rectangle is also the diameter of the semicircle, the height is equal to the radius of the semicircle:
height = r
Now, we can express the area in terms of x and r:
Area = x * r + (1/2) * π * r^2
Next, we need to find the dimensions that maximize the area. To do this, we need to find the critical points of the area function.
To find the critical points, we need to take the partial derivatives of the area function with respect to x and r and set them equal to zero.
∂Area/∂x = r = 0
∂Area/∂r = x + π * r = 0
From the first equation, we have r = 0.
Substituting r = 0 into the second equation, we have x + π * 0 = 0.
This implies x = 0.
However, since we are dealing with physical dimensions, x and r must be positive.
Therefore, the critical point (x, r) = (0, 0) is not valid.
To find the dimensions that allow the window to admit the most light, we need to consider the boundary conditions.
The area function is defined for 0 < x and 0 < r.
Let's solve the equation for the perimeter to express r in terms of x:
24 = 2x + πr + 2r
Combining like terms, we have:
24 = (2 + π) r + 2x
Simplifying, we get:
r = (24 - 2x) / (2 + π)
Since r must be positive, we have the condition:
0 < (24 - 2x) / (2 + π)
Simplifying, we have:
0 < 24 - 2x
2x < 24
x < 12
So, we have the condition:
0 < x < 12
Now, let's substitute the expression for r into the area function:
Area = x * r + (1/2) * π * r^2
Area = x * [(24 - 2x) / (2 + π)] + (1/2) * π * [(24 - 2x) / (2 + π)]^2
Next, we can simplify the area function:
Area = (x * (24 - 2x) / (2 + π)) + (π * (24 - 2x)^2 / (4 + 2π))
To find the dimensions that maximize the area, we need to find the maximum of this function for 0 < x < 12.
To find the dimensions that will allow the window to admit the most light, we need to determine the dimensions of both the rectangular part and the semicircular part.
Let's assume the rectangular part has width "w" and height "h". The semicircular part has a radius equal to half the width of the rectangular part, which is w/2.
To calculate the perimeter of the Norman window, we need to consider the perimeter of both the rectangular part and the semicircular part.
The perimeter of the rectangular part is given by: P_rectangular = 2w + 2h.
The perimeter of the semicircular part is the length of the circumference of the semicircle, which is half of the circumference of a full circle with radius w/2. Therefore, the perimeter of the semicircular part is given by: P_semicircular = (πw)/2.
The total perimeter of the Norman window is the sum of the perimeters of the rectangular and semicircular parts: P_total = P_rectangular + P_semicircular = 2w + 2h + (πw)/2.
Given that the total perimeter is 24 ft, we can write the equation:
2w + 2h + (πw)/2 = 24.
Now, in order to maximize the area of the window (which allows the most light), we need to set up an equation for the area and find the maximum value. The area of the window is equal to the sum of the area of the rectangular part and half the area of the semicircular part.
The area of the rectangular part is given by: A_rectangular = w * h.
The area of the semicircular part is half the area of a full circle with radius w/2, which is: A_semicircular = (π(w/2)^2)/2.
The total area of the window is: A_total = A_rectangular + A_semicircular = w * h + (π(w/2)^2)/2.
To find the dimensions that will allow the window to admit the most light, we need to maximize the total area A_total. We can do this by taking the derivative of A_total with respect to either w or h and setting it equal to zero to find the critical points.
Taking the derivative of A_total with respect to w, we get: dA_total/dw = h + (πw)/4.
Setting dA_total/dw equal to zero, we have: h + (πw)/4 = 0.
Solving this equation for h, we get: h = -(πw)/4.
Substituting this value of h into the equation of the total perimeter, we have:
2w + 2(-(πw)/4) + (πw)/2 = 24.
Simplifying this equation, we get: 2w - (πw)/2 = 12.
Combining like terms, we have: (4w - πw)/2 = 12.
Simplifying further, we get: (4 - π/2)w = 24.
Now, solving this equation for w, we have: w = 24 / (4 - π/2).
Once we have the value of w, we can substitute it back into the equation for h to find its value.
Finally, to find the dimensions that will allow the window to admit the most light, we need to calculate the area A_total using the values of w and h.
To summarize:
1. Set up the equation for the total perimeter using the expressions for the rectangular and semicircular parts.
2. Take the derivative of the total area with respect to one of the dimensions (w or h).
3. Solve the resulting equation for the critical points.
4. Substitute the value of one dimension into the equation for the other dimension.
5. Calculate the total area using the values of both dimensions.
6. The values of w and h that maximize the total area will be the dimensions that allow the window to admit the most light.