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 45 feet?
so the perimeter is:
pi*r + 2r + 2h = 45.
h= (45 - PIr - 2r)/2
The area equals = (pi*r^2)/2 + 2rh
what do i do from here?
<<what do i do from here?>>
(1) Substitute your h(r) equation into the A (r,h) equation to express Area (A) in terms of r only.
(2) Then compute dA/dr and set it equal to zero.
(3) The solution will be the maximum-area value of r.
(4) Then substitute that r into the A(r) equation to get the maximum area.
To find the maximum area of the Norman window, let's substitute the equation for h into the equation for the area (A).
We have: h = (45 - πr - 2r)/2
Substituting h into the equation for the area, we get:
A = (πr^2)/2 + 2r((45 - πr - 2r)/2)
Simplifying this expression:
A = (πr^2)/2 + 2r(45 - πr - 2r)/2
A = (πr^2)/2 + r(45 - πr - 2r)
Now, let's compute dA/dr, which represents the derivative of A with respect to r:
dA/dr = d(πr^2)/2/dr + d(r(45 - πr - 2r))/dr
Using the power rule of derivatives and the product rule, we can simplify this expression further:
dA/dr = (π/2)(2r) + (45 - πr - 2r) + r(-π - 4)
dA/dr = πr + 45 - πr - 2r - πr - 4r
dA/dr = 45 - 7r - 2πr
To find the maximum area, we need to set dA/dr equal to zero and solve for r:
45 - 7r - 2πr = 0
Combine like terms:
45 - (7 + 2π)r = 0
Solving for r:
r = 45/(7 + 2π)
Now that we have the value of r, we can substitute it back into the equation for the area to find the maximum area:
A = (πr^2)/2 + r(45 - πr - 2r)
Substitute r = 45/(7 + 2π) into the equation for A to find the maximum area.