Step-by-Step Solution

Topic: Maxima & Minima

Visual Representation

The window consists of a rectangle (dimensions x, y) surmounted by an equilateral triangle (side x).

(I) RELATION BETWEEN x AND y

The “Perimeter of the window” implies the outer boundary length.

Boundary = (Base) + (Left side) + (Right side) + (Two sides of the equilateral triangle).

Sides of equilateral triangle = x.

Perimeter (P) = x + y + y + x + x = 3x + 2y

Given P = 12m:

(II) AREA FUNCTION A(x)

Area = Area of Rectangle + Area of Equilateral Triangle.

Area = xy + (√3 / 4)x²

From (i), y = (12 – 3x) / 2

Substitute y into the Area equation:

A(x) = x [ (12 – 3x)/2 ] + (√3 / 4)x²

(III)(A) MAXIMIZING LIGHT (AREA)

Differentiate A(x) with respect to x:

A'(x) = 6 – 3x + (√3 / 2)x

For critical points, set A'(x) = 0:

6 – x(3 – √3/2) = 0

6 = x( (6 – √3) / 2 )

x = 12 / (6 – √3)

(III)(B) PERIMETER EXPRESSION GIVEN AREA = 50

Step 1: Use Area condition to find y

Given Area = 50 m²

Area = xy + (√3 / 4)x² = 50

xy = 50 – (√3 / 4)x²

y = (50/x) – (√3 / 4)x

Step 2: Substitute y into Perimeter formula

Perimeter P = 3x + 2y

P(x) = 3x + 2 [ (50/x) – (√3 / 4)x ]

P(x) = 3x + (100/x) – (√3 / 2)x

Combine x terms: