$PQR$ is an isosceles triangle with $PQ = PR = 50\text{cm}$ and $QR = 28\text{cm}$ as shown in

Question

$PQR$ is an isosceles triangle with $PQ = PR = 50\text{cm}$ and $QR = 28\text{cm}$ as shown in the diagram above. A rectangle $ABCD$ is drawn inside the triangle with $AB$ on $QR$, and $D$ and $C$ on $PQ$ and $PR$ respectively. If $AB = 2x\text{cm}$, show that the area $A \text{cm}^2$ of the rectangle is given by $A = \frac{48x(14-x)}{7}$. Obtain an expression for $\frac{\mathrm{d}A}{\mathrm{d}x}$. Given that $x$ can vary, find the value of $x$ for which the area of the rectangle is stationary. Explain why this value of $x$ gives the largest area possible.

Accepted Answer

(i) $A=\frac{48x(14-x)}{7}$ (ii) $\frac{\mathrm dA}{\mathrm dx}=96-\frac{96x}{7}$ (iii) $x=7\text{ cm}$ (iv) Maximum since $\frac{\mathrm d^2A}{\mathrm dx^2}=-\frac{96}{7}<0$

2017 CHIJ AM SA1 P1

2017 CHIJ AM SA1 P1

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