Fancy Window
Ok, how many began figuring out the 68 inch string problem? Ok, it was a joke. Now for the real problem. (change the circle to a semi-circle with a diameter equal to one side of a rectangle)
Joan's house has a window in the shape of a rectangle surmounted with a semicircle. For a given total perimeter of 268 inches, what are the dimensions of the window if it allows the maximum amount of light?
Hint: Remember the 68 inch string solution?
Joan's house has a window in the shape of a rectangle surmounted with a semicircle. For a given total perimeter of 268 inches, what are the dimensions of the window if it allows the maximum amount of light?
Hint: Remember the 68 inch string solution?
Labels: mathschallenge
posted by Ragknot at
5:30 PM
10 Comments:
let x be the length of the bottom
let y be the side of the rectangle part
the area will be
A = pi*x^2/8+xy
and the perimeter is
268 = x+pi*x+2y
solve for y in terms of x
268 = x+pi*x+2y
y = (268-x-pi*x)/2
substitute y in the area formula
A = pi*x^2/8+x(268-x-pi*x)/2
A = x^2*(-3pi-4)/8 + 134x
take the derivative and solve for when dA/dx=0
A = x^2*(-3pi-4)/8 + 134x
0 = x(-3pi-4)/4 + 134
x = 536/(3pi+4)
substitute it back into the perimeter formula
268 = x+pi*x+2y
268 = (536/(3pi+4))(pi+1) + 2y
y = (134pi+268)/(3pi+4)
x=39.9926
y=51.3210
To Sam:
If I understand your dimensions the
area is 2680.5463 and the perimeter is only 205.4548
Something seems amiss here.
Sam could have saved himself a lot of work if he figured that a circle always encloses a larger area than a square for the same perimeter. Same goes for a semi-circle versus an open square.
So the maximum area window will be a semi-circle with no rectangular extension below.
Sam had the right approach but an error in his perimter formula. It should be
268 = x + pi*x/2 +2y
Which gives the result y~=37.5 and x ~=(2*37.5) = 75
Maximize the circle!
This is the clue from the 68 inch string.
Draw a 37.5 inch radius circle, from the left and right quad points draw down to interset a horizontal line thru the bottom quad.
That's almost the correct figure, actually it should be slightly larger than 37.5.
The rectangular area is a square
to maximize that part.
4*x + pi*x =268
x=37.52664329
Opps, the bottom is 1/2 of the square that would extend to the top. A "square" with filleted corners.
Meh. I get part marks at least.
To Sam:
Your work is very good.
Someone said your error was in the perimeter equation. But it appears to me like where you said
A = pi*x^2/8+xy,
it should have been
A = pi*x^2/2+xy
Can you verify?
To Sam:
Your work is very good.
Someone said your error was in the perimeter equation. But it appears to me like where you said
A = pi*x^2/8+xy,
it should have been
A = pi*x^2/2+xy
Can you verify?
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