Minimizing Total Area: Optimal Dimensions for Triangles and Squares

[mattxr250]

I was wondering if someone could workout this problem...

The sum of the perimeters of an equilateral triangle and square is 10. Find the dimensions of the triangle and the square that produce a minimum total area.

Thanks for any help

 

[EnumaElish]

So you have 3L + 4L = 7L = 10, so L = 10/7, is that right? Or is it more complicated?

 

[amcavoy]

Well it could be 3L + 4W = 10. Are the sides of the triangle the same as the square?

 

[EnumaElish]

Let's see how mattxr250 thinks it is. mattxr250 are you there?

 

[HallsofIvy]

EnumaElish, If the triangle and square had the same length sides, then the side length would have to be 10/7 and there wouldn't be any question about which lengths give the minimum total area would there?

mattxr250, as apmcavoy says, the total perimeter would be 3L + 4W = 10 (I assume he means L as the length of a side of the triangle and W as the side of the square. The area of the square would be W² (I'm happy to do the easy part for you! What would the area of the triangle be in terms of L? What would the total area be? Can you put that only in terms of L (or W)? Can you find the value that makes that a minimum?

 

[mattxr250]

HallsofIvy, I had the 3L + 4W = 10...As you stated, the sides of the triangle cannot be the same length as the square because you don't have to differentiate to find a minimum area...heres what i came up with for setting that equation equal to L...

L = (10-4W)/3...

so does this make sense for the perimeter?

3[(10-4W)/3] + 4W = 10...after that I'm lost, lol...any more help?

 

[EnumaElish]

so does this make sense for the perimeter?

3[(10-4W)/3] + 4W = 10...

You have two unknowns L and W but one equation. Can you determine 2 unknowns from a single equation?

The thread title said calc optimization, so where do you think is the calculus or the optimization part? Okay, I see that you said area needs to be minimized. So how do you write the total area in terms of L and W?

 

[mattxr250]

well, as stated the area of the square is just W^2...and although the formula for the area of triangle is (1/2)b(h), I don't know how to get to that...

to get the optimization part you need the equations for the area, differentiate them, and then find the min of f '(x), but I don't know how...help??

 

[EnumaElish]

You are right about the square. How do you get from the side of an equilateral triangle to its height? I don't remember the formula for it. (The base is just L so b = L.)

 

[mattxr250]

oh, isn't that a "30, 60, 90" triangle if you draw an altitude from a vertex to the opposite side?? ok, maybe I'll try that...but...

what do I differentiate?? the two forumlas for area?

 

[mattxr250]

well, i found the formula for the area of an equilateral triangle...

[(3^(1/2))/4](L^2)...

i really need help guys...any suggestions?

 

[EnumaElish]

Okay, so you have the formulas for the triangle and the square, so you can represent total area in terms of L and W: Triangle area = T(L), Square area = S(W), so total area = T(L)+S(W). YOu had said you'd need to differentiate, and that's correct. You need to differentiate with resp. to L and W separately and set each derivative = 0. But you also know that L and W have to satisfy the condition "total circumference = 10." This last condition makes the problem a constrained optimization problem. Have you covered constrained optimization in class? Have you encountered or solved constrained optimization examples?