What sizes will give the largest window?

[tangents]

Hey Guys, I managed to just scavenge this site from google and am in need for some assitance with this problem, which i think is very straightfoward but not sure how to approach it.

A rectangular window surmounted by an equilateral triangle has a perimeter of 12ft, find the dimensions of rectangle that will give largets area of window.

Now I know for these types of question, a diagram is useful. Also I need to find a primary and secondary equation, given that if there are more than one variable. I then find the dervative and solve for 0 and do sign chart. but I am not sure if the window is inscibed in the trianlge or vice versa. Do I use the equation of the perimter 2x+2y=12 and Area which iis A: XY and plug in and find derivative? I don't see what role the trianlge plays if any.

 

[andrewchang]

it seems to me like the window is mounted inside the triangle. so then you want to find the rectangle of largest area that can be inscribed inside the triangle.

 

[TD]

English isn't my native language, but from what I understand you have a rectangle and on top of that (on the top side), a triangle.

 

[andrewchang]

actually, i think TD is right. sorry about that.

 

[TD]

In that case, construct two functions:
- the area-function (add the areas of the triangle & rectangle) which is the function you wish to maximize.
- the perimeter-function (add two sides of the triangle and the bottom + left & right side of the rectangle) which has a constant value.

Use the second function to eliminate one of the two variables in the first function so you can just take its derivative to find the maximum.

 

[andrewchang]

and you may want to take the second derivative to verify that it is, indeed, a maximum.

 

[tangents]

Ok Area of the equilateral triangle:
Rad(3)/4*x^2 and the rectangle is A=XY
Perimater is: 3x+2y=12
y=(12-3x)/2

So total area: Rad (3)/4*x^2+ (12x-3x^2)/2
Derivative: Rad(3)/2*x+6-3x. x=0 but that can't be right since that would give an area of zero ><

 

[TD]

Derivative: Rad(3)/2*x+6-3x. x=0 but that can't be right since that would give an area of zero ><

If I understand, you found the derivative to be $\sqrt 3 x/2 + 6 - 3x$. But x = 0 isn't a solution of the equation you get by setting this derivative 0.

 

[andrewchang]

Area of Rectangle: $A=xy$
Area of Triangle: $A= \frac {1}{2} bh= \frac {1}{2} (x)(\frac {\sqrt{3}x}{2})$

since you got that, your answer should come out right. check your calculations?

 

[tangents]

ah yes i got it now, i just graphed it and got the zero of the derivative, thanks you for your help guys!

 

[Unco]

The question is perhaps a little confusing.

A rectangular window surmounted by an equilateral triangle has a perimeter of 12ft, find the dimensions of rectangle that will give largest area of window.

The way I read it:

The shape given by a rectangle (the window) with an equilateral triangle on it has a perimeter given by 3x+2y=12.

The window is rectangular, so the area of the window is x*y.

3x + 2y = 12

A = xy

Am I mistaken?

 

[andrewchang]

no, youre forgetting that the triangle is also part of the window

 

[Unco]

The window is rectangular.

 

[andrewchang]

yes, the bottom part is, but its also "surmounted by an equilateral triangle"

i'm assuming the triangle is part of the window. otherwise, it makes no sense to even bother giving the information.

 

[Unco]

yes, the bottom part is, but its also "surmounted by an equilateral triangle"

The window cannot be both rectangular and pentagonal.

i'm assuming the triangle is part of the window. otherwise, it makes no sense to even bother giving the information.

Refer to my first post in the thread; the perimeter of the pentagon relates x and y, in my opinion.

Ultimately, the question is poorly-worded; no interpretation can be correct.