SOLUTION: A window is to be constructed in the shape of an equilateral triangle on top of a rectangle if its perimeter is to be 600 cm, What is the maximum possible area of the window?
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Question 1146864: A window is to be constructed in the shape of an equilateral triangle on top of a rectangle if its perimeter is to be 600 cm, What is the maximum possible area of the window?
Let W be the width of the window, in centimeters.
Then the vertical length of the rectangular part of the window is L = 0.5*(600 - 3W) centimeters.
Thus the area of this special form window is
A(w) = W*0.5*(600-3W) + cm^2. (1)
The first addend in the formula is the area of the rectangular part, while the second addend is the area of the triangular part.
Thus the area is this quadratic function of the variable "w"
A(w) = + + = + 300*W. (2)
They ask to find the maximum of this quadratic form.
For the general quadratic form f(x) = ax^2 + bx + c with negative leading coefficient "a",
the maximum is achieved at x = .
In our case, a = , b = 300.
Therefore, the quadratic form achieves the maximum at
W = = 140.583 cm.
To find the maximum area, simply substitute this value of W into the formula (2)
= + 300*140.583 = 21087.41 cm^2. ANSWER
