Question

A special window has the shape of a rectangle surmounted by an equilateral triangle. If the perimeter of the window is 16 ft, what dimensions will admit the most light?

Answer 1

@hartnn

Answer 2

i dont know where to start..if you get me started with an equation i can do this : )

Answer 3

whats the criteria for admitting most light ?

Answer 4

like max. area or something ?

Answer 5

yup

Answer 6

what dimensions give the max area is what this is asking

Answer 7

so let the length be = x
what is width = ?
from perimeter formula....

Answer 8

If x is the length of the rectangle (and side of the triangle), let height of rectangle be h.
Then the perimeter of the figure is 3x+2h=16 =>h=8-3x/2

The area of the figure, A, is given by
A=xh+(sqrt(3)/4)x^2
=x(8-3x/2)+(sqrt(3)/4)x^2

Differentiating gives
dA/dx=8-3x+(sqrt(3)/2)x
d^2A/dx^2=-3+(sqrt(3)/2)<0 =>Maximum

So maximum area of window is given when
8-x(3-(sqrt(3)/2))=0
=>x=3.75 (2dp)
=>h=2.38 (2dp)
ie the window is of length 3.75 feet and width 2.38 feet, and the equilateral triangle has sides of 3.75 feet.

Answer 9

im confused...a lot

Answer 10

the perimeter formula for the square will be 2x+2y=16

Answer 11

but thats the perimeter of entire window, 16 ft