URGENT!!!! optimization question PLZ?

Question

a picture window is in the shape of an equilateral triangle. Each side measures 4m. Celia will glue a rectangular piece of stained glass on the window so that one side of the rectangle lies on the base of the triangle. Determine the max area for the piece of stained glass.

the answer should be 2squareroot3

plz show all work :) Thank u

Answer 1

Draw the triangle with a rectangle inside it, sharing (part of) the base. Let the height of the rectangle be h m.

Consider the right-angle triangle which is the corner of the large triangle being cut off by the rectangle. This has an angle of 60° opposite a side of length h. Call the adjacent side x. Then tan 60° = √3 = h/x. So x = h/√3.

Now x is the width of the base cut off at each corner. So the width of the rectangle is 4 - 2x = 4 - 2h/√3. So the area is A = h(4 - 2h/√3) = 4h - 2h^2/√3.

Since this is a quadratic in h we know it will have a maximum value at h = -4 / (-4/√3) = √3. Then the maximum area is 4√3 - 2√3 = 2√3.

Answer 2

A good approach is to draw a picture and then express the base b of the rectangle as a function of the height h (or vice versa). Then the area is bh, you can plug in for b in terms of h and differentiate with respect to h.

Let's try it:
My picture indicates that b = 2(2 - h/sqrt(3))

So the area would be A= bh = 4h - (2/sqrt(3))h^2

dA/dh = 4 - (4/sqrt(3))h

Setting this to 0, we get h = sqrt(3).

Plugging this into the area equation, we get A = 2sqrt(3)