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 20 ft, find the dimensions of the window so that the greatest possible amount of light is admitted.
My base length is correct: (10/(1+pi/4))
But I need the total height, thanks so much!!
To find the total height of the Norman window, we need to consider the dimensions of both the rectangle and the semicircle.
Let's start by assigning variables to the dimensions of the window. Let W be the width of the rectangle (which is also the diameter of the semicircle), and let H be the total height of the window.
The perimeter of the window is given as 20 ft, which can be expressed as:
perimeter = 2 * (length of rectangle) + circumference of semicircle
From the given information, the length of the rectangle is also equal to the base length, which is 10/(1+pi/4). We can substitute this value into the equation:
20 = 2 * (10/(1+pi/4)) + circumference of semicircle
Now, let's calculate the circumference of the semicircle. The circumference of a circle is given by the formula C = π * D, where C is the circumference and D is the diameter. In this case, the diameter equals the width of the rectangle, which is W. So we have:
circumference of semicircle = (π/2) * W
Substituting this back into the equation, we get:
20 = 2 * (10/(1+pi/4)) + (π/2) * W
Simplifying further, we can solve for W:
W = (20 - 4 * (10/(1+pi/4))) * (2/π)
Now that we have the width W, we can find the total height H. Since the diameter of the semicircle is equal to the width of the rectangle, the radius of the semicircle is (1/2)W. Thus, the total height is the sum of the width and the radius:
H = W + (1/2)W
Simplifying this expression, we get:
H = (3/2)W
Now you can substitute the value of W you obtained earlier into this equation to find the total height H.
To find the total height of the Norman window, we need to consider the rectangle and the semicircle separately.
Let's denote the width of the rectangle as "w" and the base length of the rectangle as "b".
Given that the perimeter of the window is 20 ft, we can form an equation using the perimeter formula for a rectangle:
Perimeter of rectangle = 2 * (length + width)
Since the base length of the rectangle is equal to the diameter of the semicircle, which is also the width of the rectangle, we have:
2 * (b + w) = 20
Simplifying this equation, we get:
b + w = 10 ----(Equation 1)
Now, let's consider the semicircle. The circumference (perimeter) of a semicircle is half of that of a full circle:
Circumference of semicircle = π * d / 2
Since the diameter of the semicircle is equal to the width of the rectangle:
Circumference of semicircle = π * w
However, we only want to consider the curved part of the semicircle, so we need to subtract the diameter (width of the rectangle) from the total height. Therefore, the total height of the window is:
Total height (h) = Circumference of semicircle - w
= π * w - w ----(Equation 2)
To maximize the amount of light admitted, we want to maximize the area of the window. The area of the window is the combined area of the rectangle and the semicircle:
Area of window = Area of rectangle + Area of semicircle
= b * w + (π * w^2) / 2
Now, we can express the area in terms of a single variable w by substituting the value of b from Equation 1:
Area of window = w * (10 - w) + (π * w^2) / 2
To find the maximum value of this area, we can differentiate it with respect to w and set the derivative equal to zero:
d(Area of window) / dw = 0
Differentiating the area expression, we get:
10 - 2w + πw - πw / 2 = 0
Combining like terms, we have:
(10 - 2w) + (πw - πw/2) = 0
10 - 2w + πw(1 - 1/2) = 0
10 - 2w + πw/2 = 0
10 + πw/2 - 2w = 0
Now, solve this equation for w to find the value that maximizes the area.
It might be easier if the width of the rectangle is 2x and the height is y, so the perimeter is
2x+2y+πx = 20
y = 10 - (π+2)/2 x
Then the area is
A = π/2 x^2 + 2xy
= π/2 x^2 + 2x(10 - (π+2)/2 x)
= π/2 x^2 + 20x - (π+2)x^2
= -(2 + π/2)x^2 + 20x
The vertex of this parabola is at
x = 20/(4+π)
Note that our answers agree, allowing for the fact that I used 2x for the diameter.
So, y = 10 - (π+2)/2 * 10/(4+π) = 5 + 10/(4+π)