An architect uses $40$ cm of wire to design $2$ window frames for a room model. He wants as much

Question

An architect uses $40$ cm of wire to design $2$ window frames for a room model. He wants as much light as possible to pass through the windows for his design. One frame is to be an equilateral triangle of side $x$ cm. The other is to be a semi-circle of radius $r$ cm. Show that the total area, A $cm^2$, enclosed by the two window frames is given by $A = \frac{\pi(40-3x)^2}{2(\pi+2)^2}+\frac{\sqrt 3x^2}{4}$[4] Given that $x$ can vary, find the stationary value of $A$.[4] Find the nature of this stationary value and explain why the architect may not be happy with his design.[2]

Accepted Answer

(a) $A=\dfrac{\pi(40-3x)^2}{2(\pi+2)^2}+\dfrac{\sqrt{3}x^2}{4}$ (b) $42.5\ \text{cm}^2$ (c) Minimum; hence it does not give the greatest possible light.

2025 CGSS PRELIMS P1 Q8

2025 CGSS PRELIMS P1 Q8

Solution:

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