Probem59+Sabrina

Set this up -- what is your objective function? What is your "other" information? You might want to draw a diagram and scan it in to illustrate this...these are difficult problems to do without a diagram.

Isosceles triangle: A= ½bh

Equilateral Triangle: A = ½b (½b√3)

Show that of all the isosceles triangles with a given perimeter, the one with the greatest area is equilateral.

A(isos)= ½bh A(equi)= ½b (½b√3)

P(isos)= 2s + b P(equi)= 3b


 * AM I ALLOWED TO SUBSTITUTE IN REAL NUMBERS?no. And keep in mind that this is a problem in which you are looking to find an absolute extrema -- so state it: are you trying to maximize or minimize something? What is it that you are trying to maximize or minimize? What is a formula for this? Draw a diagram with your variables, and use the relationship to write your objective function as a function of one variable. This works out nicely!

I'm trying to maximize the area. But there are so many equations and different variables! i don't even know where to begin. On paper, I've written out all the different equations for the perimeters areas of equilateral and isosceles triangles.

P(isos) = 2s + b

A(isos) = 1/2 b * sqrt[s^2 + (b^2/2)] 

it's going to end up that the graph **of the area function **has an absolute extrema when s=b ok. so, using the A of the** isosceles triangle, since this is what we are trying to MAXIMIZE: s=P/3 A=1/2 b * sqrt[(P/3)^2 - ( (b^2/2)] A= 1/2 b * sqrt[P^2/9 - (b^2)/2] A'= 1/2( √P^2/9 - b^2/2) + B^2/4*(P^2/9 - b^2/2)^(-1/2)**
 * P= 3s

The Perimeter can not be completely eliminated in the derivative because it is under the sqrt symbol. 

So, I know we're trying to maximize A=1/2 b* h A= 1/2 b * sqrt(s^2 + (b/2)^2) We know that Perimeter = 2s + b P-2s=b....but we don't what P is! so I cannot for the life of me figure out how to relate s and b to one another!

