Optimizing Isosceles Triangle Area with Limited Materials

[Sethka]

Well Strange to me at least,

A man wants to build a patio on his house in the shape of an isosceles triangle. He wants to build the side walls out of pink planks, but he has only 600 yards worth of planks. Find the dimensions of the largest area he can build if he's using the side of his house as one of the walls instead of planks.

I'm not even sure where to start, all other optimization questions I've encountered where square or rectangle based. How do you go about this with a triangle?

 

[berkeman]

I'm not sure what 600 yards of planks means. Does that mean he can just make the triangle 300 yards on a side? He better have a pretty big house! Is there any info on how tall the planks are, and how the fence height comes into play?

 

[Sethka]

I don't think their height is coming into play. Sorry i was paraphrasing a bit, by planks, think of a fence, so he has 600 yards of fencing. I think i need to know how big of a triangle as seen from above he could make.

 

[berkeman]

Well unless the person has a huge house, it seems like the short side of the Isosceles triangle would have to be the side of the house. You need that dimension to figure out the area. Could you please post the exact text of the question? I have to go now, but if anybody else wants to help you, I think they will need the exact problem statement. Good luck.

http://en.wikipedia.org/wiki/Isosceles_triangle

 

[nocturnal]

As berkeman said, please post the exact statement of the problem. It is unclear if we can assume the man's house is of enough length for any triangle we desire, or if their is some kind of restriction. Also, can the planks be any two sides of the triangle?

PS - This man is building a pink fence Where is this West Hollywood?

 

[Sethka]

I'm just throwing the question out the window. I called a friend and he can't figure it out either, I think it was a bad translation or something. Here;s another one, not quiet so strange though but it's the same sort of problem.


A woman owns a large flock of sheep and she needs to build them a pen. She wants to build the pen in the shape of an isosceles triangle, with the fence making up the two equal sides of the triangle and using a nearby stream as the third side. What are the dimensions of the largest possible area that can be enclosed by 200 yards of fence?

Some of the questions I'm using to study are translated from another language into english, So I'm sorry if the word questions sound a little muddled.

 

[loom91]

I'm just throwing the question out the window. I called a friend and he can't figure it out either, I think it was a bad translation or something. Here;s another one, not quiet so strange though but it's the same sort of problem.


A woman owns a large flock of sheep and she needs to build them a pen. She wants to build the pen in the shape of an isosceles triangle, with the fence making up the two equal sides of the triangle and using a nearby stream as the third side. What are the dimensions of the largest possible area that can be enclosed by 200 yards of fence?

Some of the questions I'm using to study are translated from another language into english, So I'm sorry if the word questions sound a little muddled.

Let's see, this is a new type for me.

Let the sides be x, x and y. We have $2x = 200 => x = 100$.

The area of an isosceles triangle is given by $$\frac {1}{2} y \sqrt {x^2 - \frac {y^2}{4}}$$

Differentiating w.r.t. y, we have

$$\frac {dA}{dy} = \frac {1}{2} \sqrt {x^2 - \frac {y^2}{4}} + \frac{1}{2} y \frac {1}{2 \sqrt {x^2 - \frac {y^2}{4}}} \frac {-2y}{4}$$

Set this equal to 0 and solve for y, check for negative curvature and you are all optimized.