Optimization

[DistantBeliefs]

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 Im 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.

 

[Mrspi]

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 Im 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.

I agree; a diagram is extremely helpful in this type of problem. Draw a rectangle. Use one side of the rectangle as a side of the equilateral triangle (since the rectangle is "surmounted" by the triangle, the triangle adjoins the top of the rectangle).

Let x be the length of a side of the equilateral triangle (so, a side of the rectangle will also be x). Let y be the length of the side of the rectangle which is not shared with the triangle.

The perimeter of the window is 12. So,
12 = 3x + 2y (do you see why you need 3x?)
You can solve that to get y in terms of x.

The area of an equilateral triangle of side x is
Area = [sqrt(3)) x^2]/4

You know the formula for the area of a rectangle, I assume; your rectangle has length and width of x and y, and you should have an expression to use for y.

Area of window = area of triangle + area of rectangle

Substitute the expressions you've come up with for the area of the triangle and rectangle. Find the derivative and set it equal to 0.....solve for x. Once you have x, you can also find y.

 

[DistantBeliefs]

12 = 3x + 2y (do you see why you need 3x?)

I know that you added another side of the triangle but Im not sure why you have to do that if only one side is beign shared with the rectangle. anyway i get this:

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 cant be right since that would give an area of zero ><

 

[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 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?

 

[galactus]

I would think you have to include the triangle's area. Anyway, here's my approach:

Area of triangle is \(\displaystyle \frac{\sqrt{3}}{4}s^{2}\)

Perimeter is \(\displaystyle 3x+2y=12\)

Area of rectangles is \(\displaystyle xy\)

Total Area is \(\displaystyle xy+\frac{\sqrt{3}}{4}x^{2}\)

Solve \(\displaystyle 3x+2y=12\) for y gives \(\displaystyle y=6-\frac{3}{2}x\)

Sub into area equation:

\(\displaystyle x(6-\frac{3}{2}x)+\frac{\sqrt{3}}{2}x^{2}\)

\(\displaystyle 6x-\frac{3}{2}x^{2}+\frac{3}{4}x^{2}\)

Differentiating gives:

\(\displaystyle \frac{(\sqrt{3}-6)x}{2}+6\)

Set to 0 and solve for x gives:

\(\displaystyle \frac{4(\sqrt{3}+6)}{11}=2.81\)feet

Subbing finds \(\displaystyle y=\frac{-6(\sqrt{3}-5)}{11}=1.78 ft.\)

Does anyone concur or did I make a boo-boo?.

 

[Unco]

The window is rectangular, not pentagonal.

 

[galactus]

Maybe I am mis-interpeting the problem. I thought surmount meant to "lie at the top of". Can we just use xy and 3x+2y=12?. I suppose you're right. That would give x=2 and y=3. I am maxing the whole thing, triangle and all.

 

[Unco]

It's just a pain of a question, really. Tangents will have learned something either way, hopefully.

 

[pka]

Grammatically the sentence should be read:
“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.”
This is a poorly put problem!

 

[Unco]

That was obvious; we were trying to come up with the best guess.

 

[galactus]

It would appear someone left their participle dangle.

 

[liasia]

Re:

12 = 3x + 2y (do you see why you need 3x?)

I know that you added another side of the triangle but Im not sure why you have to do that if only one side is beign shared with the rectangle. anyway i get this:

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 cant be right since that would give an area of zero ><

i do not get this at all poop