precalc question?

Question

ive been trying forever to figure out this problem and id really appreciate some help.

"A rectangle inscribed in an equilateral triangle with a perimeter of 30 cm.

a) express area 'A' of the rectangle as a function of the height 'H' cm of the rectangle.

b) Find the dimensions of the rectangle with the largest area."

Answer 1

Since thre perimeter is given, then

30 = 2H + 2L
L = 15 - H

A = LH, so
A = (15 - H)H or
A = 15H - H^2

Since the area function is quadratic in nature, then the maximum is at the vertex of the parabola.

V(h,k) where

h = -b/2a, and
k = (4ac-b^2)/4a

Since we are only interested in the dimensions of the rectangle, getting h (which in the area function represents H) we have

h = -15/2(1)
h = 7.5, which should also be the height of your rectangle.

L = 15 - H
L = 15 - 7.5
L = 7.5

So the rectangle has to have dimensions of 7.5 by 7.5. (Note that this means the rectangle must be a square)

Hope that helps!!!

Answer 2

The triangle is equilateral with perimeter of 30 cm. So each side is length 30/3 = 10 cm.

Drop a perpendicular line from the vertex of the triangle to the base. This will divide the equilateral triangle into two congruent right triangles. They are 30-60-90 triangles. The height of those triangles is:

(10/2)√3 = 5√3 cm

Let
2x = length rectangle
h = height rectangle

The half rectangle in each triangle is
x by h

By similar triangles we have:

5√3 / 5 = h / (5 - x)
√3 = h / (5 - x)
h = √3(5 - x) = 5√3 - x√3
x√3 = 5√3 - h

x = 5 - h/√3

So the dimensions of the full rectangle is:

A = 2xh = 2(5 - h/√3)h = 10h - 2h²/√3
___________

The maximum area of the rectangle can be found by finding the vertex of the parabola.

A = 10h - 2h²/√3
A = (-2/√3)(h² - h5√3)
A + (-2/√3)(5√3/2)² = (-2/√3)[h² - h5√3 + (5√3/2)²]
A - 25√3/2 = (-2/√3)[h - 5√3/2]

The maximum area of the rectangle is

25√3/2 cm² ≈ 21.650635 cm²