Question
A window has the shape of a rectangle surmounted by an equilateral triangle. If the perimeter of the window is 12m, find the dimensions of the rectangle that will produce the largest area of the window.
✓
Answer
let sides of rectangle be x and y and the sides of equilateral triangle be x
$\therefore$3x + 2y = 12 $\Rightarrow$ $\text{y}=\frac{\text{12 - 3x}}{2}$
Area = xy + $\sqrt{3}\frac{\text{x}^{2}}{\text{4}}$
$=\text{x}\frac{\text{(12 - 3x)}}{2}+\sqrt{3}\frac{\text{x}^{2}}{4}$
$\text{A}=\frac{1}{4}[24\text{x - 6x}^{2}+\sqrt{3}\text{ x}^{2}]$
$\frac{\text{dA}}{\text{dx}}=0\Rightarrow24-12\text{x + 2}\sqrt{3}\text{ x}=0$
$\Rightarrow\text{x}=\frac{24}{12-2\sqrt{3}}\text{ OR }\frac{4(6+\sqrt{3})}{11}\text{m}$
$\therefore\text{y}=\frac{30-6\sqrt{3}}{11}\text{m}$
$\frac{\text{d}^{2}\text{A}}{\text{dx}^{2}}=(-12+3\sqrt{3})<0\therefore\text{Area is maximum for}$
$\text{x}=\frac{4(6+\sqrt{3})}{11}\text{m and y}=\frac{30-6\sqrt{3}}{11}\text{m}$.
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