My answer doesn't make sense. It's too big.
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 47 feet?
I have seen this question many times.
Your answer might be right, try sketching your diagram with the half-circle on the long side of the rectangle, if I recall that is what the shape actually looks like.
Does your answer make sense now?
No. Sorry :[
let the radius of the circle be x
then the base of the rectangle on which the circle sits is 2x
let its height be y
perimeter = 2x + 2y + 1/2(2pix)
2x+2y+pix=47
y = (47-2x-pix)/2
Area = 2xy + (1/2)pi(x^2)
= 2x(47-2x-pix)/2 + (1/2)pi(x^2)
= x(47-2x-pix) + (1/2)pi(x^2)
= 47x - 2x^2 - pix^2 + (1/2)pi(x^2)
d(Area)/dx = 47 - 4x - 2pix + pix
= 0 for a max/min of Area
solving this I got x = 6.58
so the base is 13.16
and after substituting back for y, I got
y = 6.58 as well
subbing that back in my Area equation , I got Area = 154.657
How does that match up with your answers?
My answer was around the thousands. That's how I knew it was wrong. Thanks!
To find the area of the largest possible Norman window with a perimeter of 47 feet, we first need to understand the dimensions of the window.
Let's assume that the width of the rectangle is 'x' feet. Since the diameter of the semicircle is equal to the width of the rectangle, the radius of the semicircle is 'x/2' feet.
The perimeter of the window is given as 47 feet, which can be expressed as the sum of the perimeter of the rectangle and half the circumference of the semicircle.
Perimeter = 2 * (length + width) + (π * radius)
Substituting the values and rearranging the equation:
47 = 2 * (length + x) + (π * (x/2))
Now, we can solve this equation for the length, in terms of x:
47 - (π * (x/2)) = 2 * (length + x)
23.5 - (π * (x/2)) = length + x
length = 23.5 - (π * (x/2)) - x
Now, the area of the Norman window can be calculated by multiplying the length and width of the rectangle:
Area = length * width
Since the width of the rectangle is x, the area can be expressed as:
Area = (23.5 - (π * (x/2)) - x) * x
To find the largest possible area, we can take the derivative of the area function with respect to x, set it equal to zero, and solve for x:
d(Area)/dx = 0
Once we find the value of x, we can substitute it back into the equation to find the corresponding area.