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 25 ft, find the dimensions of the window so that the greatest possible amount of light is admitted.
What is the total height?
if the rectangular window has width 2x and height h, then the area of the whole shape is
a = 2xh + pi/2 x^2
You also know that x+2h+pi*x = 25, so solve for h and plug into the formula for area. The set da/dx = 0.
To find the dimensions of the window that admit the greatest possible amount of light, we need to determine the total height of the window. Let's begin by breaking down the problem step by step:
1. Start by assigning variables to the dimensions of the window. Let's say the width of the rectangle is "w" (in feet).
2. Since the diameter of the semicircle is equal to the width of the rectangle, the radius of the semicircle will be "w/2".
3. Now, let's calculate the perimeter of the window. The perimeter of a rectangle is given by the formula: Perimeter = 2 * (length + width), and the perimeter of a semicircle is given by the formula: Perimeter = (π * radius) + 2 * radius. In this case, we have a rectangle and a semicircle, so the total perimeter is:
25 = 2 * (length + w) + (π * (w/2)) + 2 * (w/2)
4. Simplify the equation by distributing and grouping like terms:
25 = 2 * length + 2 * w + π * (w/2) + w
= 2 * length + 2 * w + (π/2)w + w
= 2 * length + (π/2 + 3)w
5. Rearrange the equation to isolate "length":
2 * length = 25 - (π/2 + 3)w
length = (25 - (π/2 + 3)w) / 2
6. We know that the height of the rectangle is equal to the radius of the semicircle, which is "w/2". So the total height is:
height = length + (w/2)
= [(25 - (π/2 + 3)w) / 2] + (w/2)
= (25 - (π/2 + 3)w + w) / 2
= (25 - (π/2 + 2)w) / 2
Therefore, the total height of the window is (25 - (π/2 + 2)w) / 2 feet.