# Area as a function

1. Sep 18, 2009

### skeeterrr

1. The problem statement, all variables and given/known data

A window has the shape of a rectangle surmounted by an equilateral triangle. Given that the perimeter of the window is 15 feet, express the area as a function of the length of one side of the equilateral triangle.

2. Relevant equations

A = lw

A = 1/2(bh)

3. The attempt at a solution

By using the Pythagorean theorem, I find the height of the triangle.

Let x represent the side length of the triangle, which is equal to the side of the rectangle which the triangle is surmounted on.

Let h represent the height of the triangle.

h^2 = x^2 - (x/2)^2

h = root(x^2 - (x/2)^2)

h = root (x^2 - (x^2)/4)

h = root ((3x^2)/4)

h = (x root(3))/2

Let P(x) represent the perimeter of the window, and let y represent the other sides that is not equal in side length of the triangle.

P(x) = 15

15 = 3x + 2y

15 - 3x = 2y

15/2 - 3/2x = y

Let A(x) represent the area of the window.

A(x) = xy + 1/2(xh)

A(x) = x(15-3x)/2 + 1/2(x(x root(3)/2)

A(x) = 15x - 3x^2 + (x^2 root(3))
---------- ------------
2 4

A(x) = 30x - 6x^2 +x^2 root(3)
------------------------
4

A(x) = x(30 - 60x + x root(3))
-----------------------
4

I am stuck here, I'm not sure if I am even doing it right... Any insights will be appreciated!

2. Sep 18, 2009

### Elucidus

Since x is the length of both the triangle sides and the sides of the square, then the perimeter consists of two triangle sides and three sides of the square (each of which is x). I'm not sure why y is needed in your formulation. I'd think you'd start with 5x = 15. Thoughts?

--Elucidus

EDIT: Nevermind. I see now the bottom portion of the window is a rectangle. I must be seeing things. I originally read "square" the first time I read the problem through.

Your work looks correct. The formula is not pretty.

3. Sep 18, 2009

### skeeterrr

Can I go further with this equation or anything? I'm a little skeptical...

4. Sep 18, 2009

### Elucidus

Only to rearrange the function expression,

e.g.

$$A(x) = \frac{(\sqrt{3} - 6)x^2 + 30x}{4}$$

or

$$A(x)=\frac{15}{2}x - \frac{3}{2}x^2 + \frac{\sqrt{3}}{4}x^2$$

or somesuch. Either way you slice it, it's clunky.

--Elucidus

5. Sep 18, 2009