A rectangle is topped by an equilateral triangle, the sides of which are equal to the width of the rectangle. Find the maximum area of the window if the perimeter is 500 cm.
so let the rectangle have
width = x
height = y
The perimeter is 2y+3x = 500
The area of the figure is
a = xy + √3/4 x^2 = x(500-3x)/2 + √3/4 x^2 = (√3-6)/4 x^2 + 250x
find where da/dx=0, or just use your knowledge of parabolas, which says that the vertex is at x = -b/2a = -250/(2 * (√3-6)/4) = 500/(6-√3)
so y = (500-3(500/(6-√3)))/2 = 250/11 (5-√3)
That makes the maximum area
(500/(6-√3))*(250/11 (5-√3)) + √3/4 (500/(6-√3))^2 = 62500/(6-√3)
Just a question, how does (√3 x^2 - 3x^2)/2 become (√3-6)/4 x^2? I'm confused
To find the maximum area of the window, we need to maximize the area of the rectangle. Let's denote the width of the rectangle as 'w' and the length of the rectangle as 'l'.
Given that the sides of the equilateral triangle on top of the rectangle are equal to the width of the rectangle, we can say that the height of the equilateral triangle is also 'w'.
The perimeter of the window is given as 500 cm. The perimeter of the window can be calculated as the sum of the perimeters of the rectangle and the equilateral triangle:
Perimeter of the rectangle = 2w + 2l
Perimeter of the equilateral triangle = 3w
Considering the given information, we can write the following equation:
2w + 2l + 3w = 500
Simplifying the equation, we get:
5w + 2l = 500
2l = 500 - 5w
l = (500 - 5w)/2
Now, let's write the formula for the area of the rectangle:
Area = length × width
Area = l × w
Area = ((500 - 5w)/2) × w
To find the maximum area, we need to find the value of 'w' that maximizes the equation for the area.
To solve the problem, we can either differentiate the area equation with respect to 'w' and set it to zero or use trial and error. Let's use trial and error in this case to find the value of 'w' that gives the maximum area.
We can start by selecting an initial value for 'w' (e.g., 50 cm). Then, we can substitute this value into the equation for the area and calculate the corresponding area.
Repeat this process for different values of 'w' and compare the areas obtained. The value of 'w' that gives the maximum area will be the answer to our problem.
For example, let's calculate the area for 'w' = 50 cm:
Area = ((500 - 5 × 50)/2) × 50
Area = (500 - 250) × 50
Area = 250 × 50
Area = 12500 cm^2
Repeat this process for different values of 'w' and observe the area obtained for each case. The largest area among the calculated values will be the maximum area of the window.