Question

If the perimeter of the window is 12 m, find the relation between x and y.

Answer 1

Total Perimeter of Composite Figure

Structure:

• Two vertical sides of the rectangle: each of length \( y \) → total: \( 2y \)
• One horizontal base of the rectangle: length \( x \)
• Two equal slant sides of the equilateral triangle on top: each of length \( x \)

Perimeter Calculation:

Total perimeter: \[ = x + 2y + 2x = 3x + 2y \]

If the total perimeter is given as 12, then: \[ 3x + 2y = 12 \quad \text{(Equation 1)} \]

Answer 2

Area of a Window (Rectangle + Equilateral Triangle)

Step 1: Expressions for Area

• Area of rectangle: \( A_1 = x \cdot y \)
• Area of equilateral triangle (side \( x \)): \[ A_2 = \frac{\sqrt{3}}{4} x^2 \]

Total area: \[ A(x) = A_1 + A_2 = x \cdot y + \frac{\sqrt{3}}{4} x^2 \]

Step 2: Use constraint from perimeter

From perimeter equation: \[ 3x + 2y = 12 \Rightarrow 2y = 12 - 3x \Rightarrow y = \frac{12 - 3x}{2} \]

Step 3: Substitute \( y \) in area formula

\[ A(x) = x \cdot \frac{12 - 3x}{2} + \frac{\sqrt{3}}{4} x^2 \] \[ A(x) = \frac{12x - 3x^2}{2} + \frac{\sqrt{3}}{4} x^2 \]

✅ Final Area Expression:

\[ \boxed{A(x) = \frac{12x - 3x^2}{2} + \frac{\sqrt{3}}{4} x^2} \]

Answer 3

Find Dimensions for Maximum Area

Step 1: Given Area Expression

\[ A(x) = \frac{12x - 3x^2}{2} + \frac{\sqrt{3}}{4}x^2 \]

Step 2: Combine Like Terms

\[ A(x) = 6x - \frac{3}{2}x^2 + \frac{\sqrt{3}}{4}x^2 = 6x + x^2\left( \frac{\sqrt{3}}{4} - \frac{3}{2} \right) \Rightarrow A(x) = 6x + x^2 \cdot \frac{\sqrt{3} - 6}{4} \]

Step 3: Differentiate to Find Maximum

\[ \frac{dA}{dx} = 6 + 2x \cdot \frac{\sqrt{3} - 6}{4} = 6 + x \cdot \frac{\sqrt{3} - 6}{2} \]

Step 4: Set Derivative = 0

\[ 6 + x \cdot \frac{\sqrt{3} - 6}{2} = 0 \Rightarrow x(\sqrt{3} - 6) = -12 \Rightarrow x = \frac{-12}{\sqrt{3} - 6} \]

Rationalize denominator (optional):

\[ x = \frac{-12}{\sqrt{3} - 6} \cdot \frac{\sqrt{3} + 6}{\sqrt{3} + 6} = \frac{-12(\sqrt{3} + 6)}{(\sqrt{3} - 6)(\sqrt{3} + 6)} = \frac{-12(\sqrt{3} + 6)}{3 - 36} = \frac{-12(\sqrt{3} + 6)}{-33} = \frac{12(\sqrt{3} + 6)}{33} \]

Step 5: Find \( y \) using constraint

From perimeter: \[ y = \frac{12 - 3x}{2} \] Substitute the value of \( x \):

\[ y = \frac{12 - 3 \cdot \left( \frac{12}{6 - \sqrt{3}} \right)}{2} \quad \text{or use the rationalized form if required} \]

✅ Final Answer:

\[ \boxed{ x = \frac{12}{6 - \sqrt{3}}, \quad y = \frac{12 - 3x}{2} } \]

These are the dimensions of the rectangle that allow maximum light through the window.

Answer 4

Expressing Perimeter \( P(x) \) as a Function of \( x \)

Step 1: Total Area of Window

The window consists of a rectangle of base \( x \) and height \( y \), and an equilateral triangle of base \( x \): \[ \text{Total Area} = \text{Area of Rectangle} + \text{Area of Triangle} = xy + \frac{\sqrt{3}}{4}x^2 \]

Given: \[ xy + \frac{\sqrt{3}}{4}x^2 = 50 \Rightarrow y = \frac{50 - \frac{\sqrt{3}}{4}x^2}{x} \]

Step 2: Total Perimeter

Perimeter includes:

• Base of rectangle: \( x \)
• Two vertical sides of rectangle: \( 2y \)
• Two slanted sides of triangle: \( 2x \)

So total perimeter: \[ P = x + 2y + 2x = 3x + 2y \] Now substitute the value of \( y \): \[ P(x) = 3x + 2 \cdot \left( \frac{50 - \frac{\sqrt{3}}{4}x^2}{x} \right) = 3x + \frac{100}{x} - \frac{\sqrt{3}}{2}x \]

✅ Final Expression:

\[ \boxed{P(x) = 3x + \frac{100}{x} - \frac{\sqrt{3}}{2}x} \]