- #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))
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2 4
A(x) = 30x - 6x^2 +x^2 root(3)
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4
A(x) = x(30 - 60x + x root(3))
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4
I am stuck here, I'm not sure if I am even doing it right... Any insights will be appreciated!