Math Help - Optimization for a triangle + square

1. Optimization for a triangle + square

1,000 feet of fencing is going to be cut into two pieces. One of the pieces will be used to enclose a square playground and the other will be to enclose an equilateral triangle playground. What should be the dimensions of the two playgrounds to maximize the area the kids have to play?

I have this so far:

Atriangle = 31/2/4 * L2
ASquare = X2
4X + 3L = 1000

Now I'm stuck; I do not know what to do. I'm very confused on how those two shapes sides can relate to one-another.

2. Re: Optimization for a triangle + square

Originally Posted by kramtastic
1,000 feet of fencing is going to be cut into two pieces. One of the pieces will be used to enclose a square playground and the other will be to enclose an equilateral triangle playground. What should be the dimensions of the two playgrounds to maximize the area the kids have to play?

I have this so far:

Atriangle = 31/2/4 * L2
ASquare = X2
4X + 3L = 1000

Now I'm stuck; I do not know what to do. I'm very confused on how those two shapes sides can relate to one-another.
So the total area is given by

$A_{\text{total}}=A_{\text{triangle}}+A_{\text{squa re}}$

So this gives

$A=\frac{\sqrt{3}}{4}L^2+x^2$

Now using the 2nd equation we can eliminate x (or L) to get an equation of one variable

$A(L)=\frac{\sqrt{3}}{4}L^2+\left( \frac{1000-3L}{4}\right)^2$

Can you finish from here?

3. Re: Optimization for a triangle + square

Originally Posted by TheEmptySet
So the total area is given by

$A_{\text{total}}=A_{\text{triangle}}+A_{\text{squa re}}$

So this gives

$A=\frac{\sqrt{3}}{4}L^2+x^2$

Now using the 2nd equation we can eliminate x (or L) to get an equation of one variable

$A(L)=\frac{\sqrt{3}}{4}L^2+\left( \frac{1000-3L}{4}\right)^2$

Can you finish from here?
Thank you so much! I believe I can finish from there, I just would like to clarify something:
From there, I find the derivative, set the derivative to 0 and solve for L, correct?

4. Re: Optimization for a triangle + square

Originally Posted by kramtastic
Thank you so much! I believe I can finish from there, I just would like to clarify something:
From there, I find the derivative, set the derivative to 0 and solve for L, correct?
Yes. That is correct.

5. Re: Optimization for a triangle + square

Originally Posted by TheEmptySet
Yes. That is correct.
Thank you! This is solved now