# 1
a special window has the shape o a rectangle surmounted by an eqilatera tiangle. ifthe perimeter o the winow is 16 ft, what dimensions will admit te most light (area of triangle= suarre root of 3/4 times x2.
To find the dimensions that will admit the most light for the given special window, we need to optimize the area of the window by maximizing the area of the equilateral triangle while ensuring that the perimeter of the window is 16 ft.
Let's denote the length of each side of the equilateral triangle as "x". Since an equilateral triangle has all sides equal, the perimeter of the triangle will be 3x.
The rectangle that forms the base of the window will have a width equal to the length of the equilateral triangle's side (x) and a length "l". The rectangle's perimeter will be 2x + 2l.
Given that the total perimeter of the window is 16 ft, we can express this mathematically as:
2x + 2l + 3x = 16
Simplifying the equation, we have:
5x + 2l = 16
To optimize the area of the window, we need to maximize the area of the equilateral triangle. The formula to calculate the area of an equilateral triangle is A = (sqrt(3) / 4) * x^2.
To find the dimensions that maximize the area, we can solve for "l" in terms of "x" using the equation for the perimeter, and then substitute that value into the area formula. Then, we can differentiate the area equation with respect to "x" and find the critical point(s) to identify the dimensions that maximize the area.
Let's differentiate the area equation:
A = (sqrt(3) / 4) * x^2
dA/dx = (sqrt(3) / 4) * 2x
To find the critical point(s), set the derivative equal to 0 and solve for "x":
(sqrt(3) / 4) * 2x = 0
2x = 0 (since sqrt(3) / 4 is a non-zero constant)
x = 0
However, x cannot be equal to 0 since it represents the length of a side. Therefore, there are no critical points.
Since there are no critical points, this means that the maximum area occurs at the endpoints of the possible range for "x". In this case, the range for "x" is limited by the perimeter equation:
5x + 2l = 16
Let's solve for "l" in terms of "x":
2l = 16 - 5x
l = (16 - 5x) / 2
To find the dimensions that admit the most light, we need to maximize the area of the window. So, let's substitute the expression for "l" into the area equation:
A = (sqrt(3) / 4) * x^2
A = (sqrt(3) / 4) * x^2 * (16 - 5x) / 2
To find the value of "x" that maximizes the area, we can differentiate the area equation with respect to "x" and find the critical point(s) again:
dA/dx = 0
By differentiating and solving for "x", you can find the value of "x" that maximizes the area. Then, substitute this value of "x" back into the perimeter equation to find the corresponding length "l". Finally, you can calculate the area of the equilateral triangle using the found values of "x" and "l".