Standard +0.3 This is a standard optimization problem requiring the constraint equation x² + y² = 16 from the semicircle, substituting to get A = 2x√(16-x²), then differentiating using the product rule. Part (b) guides students to the answer y = x, making it easier than an open-ended optimization. Slightly easier than average due to the structured parts and standard technique.
A rectangle is inscribed in a semicircle with centre \(O\) and radius 4. The point \(P ( x , y )\) is the vertex of the rectangle in the first quadrant as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{72bb1603-edbd-4e2e-bf2b-f33bb667e61b-6_553_929_1430_593}
a) Express the area \(A\) of the rectangle as a function of \(x\).
b) Show that the maximum value of \(A\) occurs when \(y = x\).
A rectangle is inscribed in a semicircle with centre $O$ and radius 4. The point $P ( x , y )$ is the vertex of the rectangle in the first quadrant as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{72bb1603-edbd-4e2e-bf2b-f33bb667e61b-6_553_929_1430_593}\\
a) Express the area $A$ of the rectangle as a function of $x$.\\
b) Show that the maximum value of $A$ occurs when $y = x$.
\hfill \mbox{\textit{WJEC Unit 3 2022 Q15}}