SPS SPS FM 2024 February — Question 1 3 marks

Exam BoardSPS
ModuleSPS FM (SPS FM)
Year2024
SessionFebruary
Marks3
TopicVolumes of Revolution
TypeVolume with implicit or parametric curves
DifficultyChallenging +1.2 This is a multi-step volumes of revolution problem requiring integration about the y-axis with different regions (cylinder, circular arc). While it involves a circle equation and requires careful setup of multiple integrals, the techniques are standard A-level Further Maths content with straightforward algebraic manipulation. The main challenge is organizational rather than conceptual.
Spec4.08d Volumes of revolution: about x and y axes
  1. O is the origin of a coordinate system whose units are cm .
The points \(A , B , C\) and \(D\) have coordinates \(( 1,0 ) , ( 1,4 ) , ( 6,9 )\) and \(( 0,9 )\) respectively.
The arc \(B C\) is part of the curve with equation \(x ^ { 2 } + ( y - 10 ) ^ { 2 } = 37\).
The closed shape \(O A B C D\) is formed, in turn, from the line segments \(O A\) and \(A B\), the arc \(B C\) and the line segments \(C D\) and \(D O\) (see diagram).
A funnel can be modelled by rotating \(O A B C D\) by \(2 \pi\) radians about the \(y\)-axis. \includegraphics[max width=\textwidth, alt={}, center]{4e1bb995-ce3d-4d16-a0a2-72383489ffe1-04_510_894_443_226} Find the volume of the funnel according to the model.
[0pt] 
\begin{enumerate}
  \item O is the origin of a coordinate system whose units are cm .
\end{enumerate}

The points $A , B , C$ and $D$ have coordinates $( 1,0 ) , ( 1,4 ) , ( 6,9 )$ and $( 0,9 )$ respectively.\\
The arc $B C$ is part of the curve with equation $x ^ { 2 } + ( y - 10 ) ^ { 2 } = 37$.\\
The closed shape $O A B C D$ is formed, in turn, from the line segments $O A$ and $A B$, the arc $B C$ and the line segments $C D$ and $D O$ (see diagram).\\
A funnel can be modelled by rotating $O A B C D$ by $2 \pi$ radians about the $y$-axis.\\
\includegraphics[max width=\textwidth, alt={}, center]{4e1bb995-ce3d-4d16-a0a2-72383489ffe1-04_510_894_443_226}

Find the volume of the funnel according to the model.\\[0pt]
\\

\hfill \mbox{\textit{SPS SPS FM 2024 Q1 [3]}}
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