Edexcel C2 2008 January — Question 8

Exam BoardEdexcel
ModuleC2 (Core Mathematics 2)
Year2008
SessionJanuary
TopicCircles
  1. A circle \(C\) has centre \(M ( 6,4 )\) and radius 3 .
    1. Write down the equation of the circle in the form
    $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = r ^ { 2 }$$ \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{13c5a854-baea-4875-82bc-86a19c3be09c-12_833_1276_605_322}
    \end{figure} Figure 3 shows the circle \(C\). The point \(T\) lies on the circle and the tangent at \(T\) passes through the point \(P ( 12,6 )\). The line \(M P\) cuts the circle at \(Q\).
  2. Show that the angle \(T M Q\) is 1.0766 radians to 4 decimal places. The shaded region \(T P Q\) is bounded by the straight lines \(T P , Q P\) and the arc \(T Q\), as shown in Figure 3.
  3. Find the area of the shaded region \(T P Q\). Give your answer to 3 decimal places. \section*{9.} \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 4} \includegraphics[alt={},max width=\textwidth]{13c5a854-baea-4875-82bc-86a19c3be09c-14_675_844_283_534}
    \end{figure} Figure 4 shows an open-topped water tank, in the shape of a cuboid, which is made of sheet metal. The base of the tank is a rectangle \(x\) metres by \(y\) metres. The height of the tank is \(x\) metres. The capacity of the tank is \(100 \mathrm {~m} ^ { 3 }\).
  4. Show that the area \(A \mathrm {~m} ^ { 2 }\) of the sheet metal used to make the tank is given by $$A = \frac { 300 } { x } + 2 x ^ { 2 }$$
  5. Use calculus to find the value of \(x\) for which \(A\) is stationary.
  6. Prove that this value of \(x\) gives a minimum value of \(A\).
  7. Calculate the minimum area of sheet metal needed to make the tank.
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