Challenging +1.8 This is a Further Maths optimization problem requiring implicit differentiation of a conic section (rotated ellipse) to find maximum y-value. Students must recognize that at maximum height dy/dx is undefined (vertical tangent), implicitly differentiate to get 2x + 2y + 2x(dy/dx) + 4y(dy/dx) = 0, set the coefficient of dy/dx to zero giving x + 2y = 0, then substitute back into the original equation to solve for y. While the technique is standard for FM, the multi-step reasoning and algebraic manipulation with the rotated conic makes this moderately challenging.
A sculpture formed from a prism is fixed on a horizontal platform, as shown in the diagram.
The shape of the cross-section of the sculpture can be modelled by the equation
\(x^2 + 2xy + 2y^2 = 10\), where \(x\) and \(y\) are measured in metres.
The \(x\) and \(y\) axes are horizontal and vertical respectively.
\includegraphics{figure_3}
Find the maximum vertical height above the platform of the sculpture.
[8 marks]
A sculpture formed from a prism is fixed on a horizontal platform, as shown in the diagram.
The shape of the cross-section of the sculpture can be modelled by the equation
$x^2 + 2xy + 2y^2 = 10$, where $x$ and $y$ are measured in metres.
The $x$ and $y$ axes are horizontal and vertical respectively.
\includegraphics{figure_3}
Find the maximum vertical height above the platform of the sculpture.
[8 marks]
\hfill \mbox{\textit{SPS SPS FM Pure 2021 Q10 [8]}}