A Norman window has the shape of a semicircle atop a rectangle so that the diameter of the semicircle is equal to the width of the rectangle. What is the area of the largest possible Norman window with a perimeter of 49 feet?
set da/dw = 0 when
w + 2h + πw/2 = 49
a = wh + π/8 w^2
= w(49 - (1+π/2)w)/2 + π/8 w^2
To find the area of the largest possible Norman window, we need to first determine the dimensions of the rectangle and semicircle that will yield the maximum area.
Let's assume that the width of the rectangle is x feet. Since the diameter of the semicircle is also equal to the width of the rectangle, the radius of the semicircle will be x/2 feet.
The perimeter of the Norman window is given as 49 feet. The perimeter of the window consists of the sum of the lengths of the four sides, which include the two sides of the rectangle and the curved part (half-circumference) of the semicircle.
The perimeter of the rectangle is the sum of the lengths of its four sides, which are all equal to the width x. Thus, the perimeter of the rectangle is 2x.
The half-circumference of the semicircle is equal to half the circumference of a circle with radius x/2, which is π(x/2). So, the perimeter of the Norman window can be expressed as:
2x + π(x/2)
Since we are given that the perimeter is 49 feet, we have the equation:
2x + π(x/2) = 49
To find the value of x, we can solve this equation.
First, let's simplify the equation:
2x + (π/2)x = 49
Next, let's combine the x terms:
(2 + π/2)x = 49
To isolate x, divide both sides of the equation by (2 + π/2):
x = 49 / (2 + π/2)
Now that we have the value of x, we can substitute it back into the equation for the perimeter to find the dimensions of the Norman window.
The width of the rectangle is x, so it is 49 / (2 + π/2) feet.
The height of the rectangle is equal to the radius of the semicircle, which is x/2. So, it is (49 / (2 + π/2))/2 feet.
Finally, we can calculate the area of the Norman window:
Area = (width of rectangle) * (height of rectangle)
Area = (49 / (2 + π/2)) * ((49 / (2 + π/2))/2)
Now, plug this equation into a calculator to find the exact value. The area of the largest possible Norman window with a perimeter of 49 feet is approximately 164.773 square feet.