Maximize the area of a kite with sides a, b and perpendicular diagonals

[nietzsche]

I just can't figure this problem out.

Homework Statement

You have four pieces of wood, two with length a and two with length b, and you arrange them in the shape of a kite (pieces of equal length placed adjacent to each other). You want to build a cross in the middle as a support. How long should the middle pieces be to maximize the area of the kite?

Sorry, there was a picture in the textbook, but it should be pretty straight forward.

Homework Equations

It's optimization.

The Attempt at a Solution

I can't figure out how to relate the variables to each other. I know that you can cut the kite in half symmetrically, and rearrange the kite into a rectangle with length y and width x. That was what I tried to do, but I still couldn't figure it out. I ended up with a whole bunch of square roots and I couldn't isolate the variable.

I tried arranging the pieces in different triangles, but I ended up with too many variables.

Any hints would be appreciated.

 

[Mark44]

Are you sure you have posted the information of this problem exactly as given? For one thing, you don't make a kite by building a frame around it. And besides, after you have used your four pieces of wood, what are you supposed to use to build the cross pieces?

For another thing, if a != b, your kite is in the shape of a rhombus; the angle formed by a short piece and a long piece is not 90 degrees. You can't cut the kite in half and form a rectangle. Instead, what you get is a parallelogram.

 

[nietzsche]

sorry, i meant cut the kite into four. then you can rearrange it into a rectangle.

i was looking at the problem in the textbook and i noticed a little symbol beside the number. it said "CAS". and so i searched the textbook for what "CAS" meant and it turns out it stands for "computer algebra system". in other words, my instructor assigned us a problem that she shouldn't have, because we don't learn how to use computer algebra systems in this course. thanks anyway.

 

[Mark44]

sorry, i meant cut the kite into four. then you can rearrange it into a rectangle.

No you can't, not of the values of a and b are different. For a rectangle, sides a and b have to meet at a 90 degree angle. This can't happen if a and b are different.

i was looking at the problem in the textbook and i noticed a little symbol beside the number. it said "CAS". and so i searched the textbook for what "CAS" meant and it turns out it stands for "computer algebra system". in other words, my instructor assigned us a problem that she shouldn't have, because we don't learn how to use computer algebra systems in this course. thanks anyway.

 

[nietzsche]

No you can't, not of the values of a and b are different. For a rectangle, sides a and b have to meet at a 90 degree angle. This can't happen if a and b are different.

 

[Mark44]

I stand corrected.

 

[nietzsche]

hehe

either way, the problem is too hard for the level I'm at. need a computer.

 

[Дьявол]

$$P = \frac{1}{2}d_1 d_2 = ab \sin(\angle ab)$$

If we represent is as function f(x)=absin(x), then to maximize the area f'(x)=0 and f''(x)>0.

So f'(x)=abcos(x) .

abcos(x)=0

cos(x)=0

And x=п/2.

f''(x)=absin(x)

f''(п/2)=ab>0 and for x=п/2 the area is maximized.