Area of Window w/ Perimeter 15ft: Function of Side Length

In summary, the perimeter of a window with a rectangular base and an equilateral triangle top can be expressed as 15 feet. Using the Pythagorean theorem and setting x as the length of the triangle side and rectangle side, the height of the triangle can be found. The area of the window can then be expressed as a function of x, with options to rearrange the function expression. However, the resulting expression is not aesthetically pleasing.
  • #1
skeeterrr
14
0

Homework Statement



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.

Homework Equations



A = lw

A = 1/2(bh)

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!
 
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  • #2
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
Thanks for the reply

Can I go further with this equation or anything? I'm a little skeptical...
 
  • #4
Only to rearrange the function expression,

e.g.

[tex]A(x) = \frac{(\sqrt{3} - 6)x^2 + 30x}{4}[/tex]

or

[tex]A(x)=\frac{15}{2}x - \frac{3}{2}x^2 + \frac{\sqrt{3}}{4}x^2[/tex]

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

--Elucidus
 
  • #5
Cool, thanks for your help!
 

1. How do you calculate the area of a window with a perimeter of 15 feet?

The formula for calculating the area of a rectangle is length x width. Since we know the perimeter is 15 feet, we can set up the equation 2(length + width) = 15. We can then solve for either the length or width, and use that value to calculate the area.

2. What is the function of the side length in determining the area of the window?

The side length is an important factor in calculating the area of the window because it determines the dimensions of the rectangle. The length and width of the window are used to calculate the area, so if the side length changes, the area will also change.

3. Is the area of the window affected by the shape of the window?

Yes, the shape of the window will affect the area. A rectangular window will have a different area than a square window with the same perimeter. This is because the side lengths will be different, resulting in a different area calculation.

4. How can the area of the window be maximized with a perimeter of 15 feet?

The area of a rectangle is maximized when the length and width are equal. In this case, the window would need to be a square with each side measuring 3.75 feet. This would result in an area of 14.06 square feet, which is the maximum area for a perimeter of 15 feet.

5. Can the function of side length be used to determine the perimeter of the window?

No, the function of side length is used to calculate the area of the window, not the perimeter. The perimeter is the distance around the window, which is determined by adding all four sides together. The side length only determines two of the four sides, so it cannot be used to find the perimeter.

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