# 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, we need to maximize the total area of the window, which is the sum of the rectangle and triangle areas. Let's break it down step-by-step:
Step 1: Define the variables
Let's denote the width of the rectangle as 'w' and the side length of the equilateral triangle as 'x'.
Step 2: Determine the perimeter equation
The given perimeter is 16 ft. Since the window consists of a rectangle and a triangle, we can express the perimeter as the sum of their side lengths.
Perimeter = 2w + 3x = 16
Step 3: Simplify the perimeter equation
Rearrange the equation to isolate 'w':
2w = 16 - 3x
w = (16 - 3x) / 2
Step 4: Calculate the area of the rectangle
The area of a rectangle is given by the formula: Area = length * width.
Since the window has a rectangular shape, the area is equal to w * x.
Step 5: Calculate the area of the triangle
The area of an equilateral triangle can be calculated using the formula: Area = (sqrt(3) / 4) * side^2.
In this case, the side length of the triangle is x. So, the area of the equilateral triangle is (sqrt(3) / 4) * x^2.
Step 6: Write the total area equation
The total area of the window is the sum of the rectangle area and the triangle area.
Total Area = w * x + (sqrt(3) / 4) * x^2.
Step 7: Substitute the value of 'w'
Replace w in the total area equation with the expression we derived in Step 3:
Total Area = [(16 - 3x) / 2] * x + (sqrt(3) / 4) * x^2.
Step 8: Simplify the total area equation
Distribute and simplify the equation:
Total Area = (16x - 3x^2) / 2 + (sqrt(3) / 4) * x^2.
Step 9: Find the derivative of the total area equation
To find the critical points (where the area is maximized), we need to find the derivative of the total area equation with respect to x and set it equal to zero.
Step 10: Differentiate the equation
Differentiate the total area equation with respect to x:
d(Total Area) / dx = (16 - 6x) / 2 + (sqrt(3) / 2) * x.
Step 11: Set the derivative equal to zero
Set the derivative equation equal to zero and solve for x:
(16 - 6x) / 2 + (sqrt(3) / 2) * x = 0.
Step 12: Solve for x
Now we can solve for x:
16 - 6x + (sqrt(3)) * x = 0.
Step 13: Simplify and isolate x
Rearrange the equation:
(3 - sqrt(3)) * x = 16.
Divide both sides by (3 - sqrt(3)):
x = 16 / (3 - sqrt(3)).
Step 14: Calculate the value of x
Evaluate the expression for x using a calculator or approximations:
x ≈ 9.83 ft.
Step 15: Calculate the value of w
Plug the value of x into the equation for w from Step 3:
w = (16 - 3 * 9.83) / 2.
Simplify:
w ≈ 0.5 ft.
Therefore, the dimensions that will admit the most light are approximately:
Width (w) ≈ 0.5 ft
Side length (x) ≈ 9.83 ft.
To find the dimensions that will admit the most light, we need to maximize the area of the window.
Let's assume that the rectangle part of the window has length "L" and width "W". The equilateral triangle on top has sides of length "x".
To start, let's express the perimeter of the window in terms of the given dimensions:
Perimeter = 2L + W + 3x
We are given that the perimeter is 16 ft, so we can write this equation:
2L + W + 3x = 16
Next, let's express the area of the equilateral triangle in terms of "x":
Area of triangle = √(3/4) * x^2
Now, the total area of the window is the sum of the area of the rectangle and the area of the equilateral triangle:
Total Area = L * W + (√(3/4) * x^2)
To find the dimensions that maximize the total area, we can use calculus. We need to find the values of L, W, and x that maximize the total area while satisfying the perimeter constraint.
The steps involved in using calculus to find the values are:
1. Solve the perimeter equation for one of the variables (e.g., L) in terms of the other variables (W and x).
2. Substitute this expression for L into the equation for the total area.
3. Find the partial derivatives of the total area with respect to W and x.
4. Set these derivatives equal to zero and solve for the values of W and x that maximize the area.
5. Check the second derivative test to ensure that the solution yields a maximum.
Since this involves algebraic manipulation and calculus, it may be more suitable to use specific numerical methods or software tools to find the exact dimensions.
However, if you are looking for an approximate solution, you can try different values of x while keeping the perimeter constraint in mind and calculate the corresponding dimensions for L and W. Then calculate the corresponding area and choose the dimensions that yield the maximum area.
let the length of the rectangle be 2x
then the sides of the equilateral triangle are also 2x
and the height of the triangle is √3 x
let the height of the rectangle be y
2y + 6x = 16
y + 3x = 8
y = 8-3x
area = 2xy + (1/2)√3 x(2x)
= 2x(8-3x) + √3 x^2
= 16x - 6x^2 + √3 x^2
d(area)/dx = 16 - 12x + 2√3 x = 0 for a max/min
16 = 12x - 2√3x)
8 = x(6 - √3)
x = 8/(6-√3)
2x = 16/(6-√3) = appr 3.75
y = appr .377
check my arithmetic