Hey Jiskha,
came across this problem in my maths homework and I can't seem to solve it.
Can someone maybe help me out?
A Norman window has the shape of a rectangle with a semi circle on top; diameter of the semicircle exactly matches the width of the rectangle. Find the dimensions w * h of the Norman window whose perimeter is 500in. that has maximal area.
Answer in inches:
w=
h=
There is also a picture with it.
i(DOT)imgur(DOT)com/kN8kmbI.png
Thanks for the help!
Maximal area of rectangle is a square.
4w = 500
Solve for w.
Unfortunately it is not that easy. I tried. :D
If you have a chance to look at the picture you will understand. :)
Just replace the "(DOT)" in the link with actual dots ".".
see the related questions below. It does get a bit messy.
To find the dimensions of the Norman window that has the maximal area, we need to use the concept of calculus and optimization.
Let's break down the problem into different parts.
1. Understand the problem:
A Norman window has the shape of a rectangle with a semicircle on top. The diameter of the semicircle matches the width of the rectangle. We need to find the dimensions of the window (width and height) that will result in the largest area.
2. Define the dimensions:
Let's assume that the width of the rectangle is 'w' inches, and the height of the rectangle is 'h' inches.
3. Set up the equation for the perimeter:
The perimeter of the window is given as 500 inches. Since the width of the rectangle is 'w' inches and the semicircle's diameter is also 'w' inches, the perimeter equation can be written as:
Perimeter = 2w + πw = 500
Simplifying, we get:
2w + πw = 500
4. Solve the equation:
To find the value of 'w', we need to isolate it in the equation. Here's how:
2w + πw = 500
w(2 + π) = 500
Dividing both sides of the equation by (2 + π):
w = 500 / (2 + π)
Now, we can substitute this value of 'w' back into the equation to find the value of 'h' (height).
5. Calculate 'h':
Since the height of the rectangle is 'h', we can express it in terms of 'w'. We know that the diameter of the semicircle is 'w', so the radius is 'w/2'. Therefore, the height 'h' is equal to the radius 'w/2'.
h = w/2
6. Substitute the value of 'w' into the height equation:
Substituting the value of 'w' into the equation, we get:
h = (500 / (2 + π)) / 2
Now, you can calculate the values of 'w' and 'h' using a calculator.
Remember, the problem asks for the dimensions 'w' and 'h' in inches.