Quadratic modelling problems

5. \begin{figure}[h] \end{figure} The share value of two companies, company \(A\) and company \(B\), has been monitored over a 15-year period. The share value \(P _ { A }\) of company \(\boldsymbol { A }\), in millions of pounds, is modelled by the equation $$P _ { A } = 53 - 0.4 ( t - 8 ) ^ { 2 } \quad t \geqslant 0$$ where \(t\) is the number of years after monitoring began. The share value \(P _ { B }\) of company \(B\), in millions of pounds, is modelled by the equation $$P _ { B } = - 1.6 t + 44.2 \quad t \geqslant 0$$ where \(t\) is the number of years after monitoring began. Figure 2 shows a graph of both models. Use the equations of one or both models to answer parts (a) to (d).

1. Find the difference between the share value of company \(\boldsymbol { A }\) and the share value of company \(\boldsymbol { B }\) at the point monitoring began.
2. State the maximum share value of company \(\boldsymbol { A }\) during the 15-year period.
3. Find, using algebra and showing your working, the times during this 15-year period when the share value of company \(\boldsymbol { A }\) was greater than the share value of company \(\boldsymbol { B }\).
4. Explain why the model for company \(\boldsymbol { A }\) should not be used to predict its share value when \(t = 20\) \includegraphics[max width=\textwidth, alt={}, center]{877d03f2-d62c-4060-bdd2-f0d5dfbe6203-17_2644_1838_121_116}

3 The curve with equation \(y = \frac { 1 } { 5 } x ( 10 - x )\) is used to model the arch of a bridge over a road, where \(x\) and \(y\) are distances in metres, with the origin as shown in Fig. 12.1. The \(x\)-axis represents the road surface. \begin{figure}[h] \end{figure}

1. State the value of \(x\) at A , where the arch meets the road.
2. Using symmetry, or otherwise, state the value of \(x\) at the maximum point B of the graph. Hence find the height of the arch.
3. Fig. 12.2 shows a lorry which is 4 m high and 3 m wide, with its cross-section modelled as a rectangle. Find the value of \(d\) when the lorry is in the centre of the road. Hence show that the lorry can pass through this arch. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{fed65420-9ef9-41d6-a58f-3b0f801d6225-3_528_870_1558_717} \captionsetup{labelformat=empty} \caption{Fig. 12.2}
   \end{figure}
4. Another lorry, also modelled as having a rectangular cross-section, has height 4.5 m and just touches the arch when it is in the centre of the road. Find the width of this lorry, giving your answer in surd form.

12 The curve with equation \(y = \frac { 1 } { 5 } x ( 10 - x )\) is used to model the arch of a bridge over a road, where \(x\) and \(y\) are distances in metres, with the origin as shown in Fig. 12.1. The \(x\)-axis represents the road surface. \begin{figure}[h] \end{figure}

1. State the value of \(x\) at A , where the arch meets the road.
2. Using symmetry, or otherwise, state the value of \(x\) at the maximum point B of the graph. Hence find the height of the arch.
3. Fig. 12.2 shows a lorry which is 4 m high and 3 m wide, with its cross-section modelled as a rectangle. Find the value of \(d\) when the lorry is in the centre of the road. Hence show that the lorry can pass through this arch. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{ede57eaa-2645-49df-aa09-68b6d5f35a9a-4_529_871_1489_678} \captionsetup{labelformat=empty} \caption{Fig. 12.2}
   \end{figure}
4. Another lorry, also modelled as having a rectangular cross-section, has height 4.5 m and just touches the arch when it is in the centre of the road. Find the width of this lorry, giving your answer in surd form.
   [0pt] [5]

8. \begin{figure}[h] \end{figure} In a competition, competitors are going to kick a ball over the barrier walls. The height of the barrier walls are each 9 metres high and 50 cm wide and stand on horizontal ground. The figure 2 is a graph showing the motion of a ball. The ball reaches a maximum height of 12 metres and hits the ground at a point 80 metres from where its kicked.
a. Find a quadratic equation linking \(Y\) with \(x\) that models this situation. The ball pass over the barrier walls.
b. Use your equation to deduce that the ball should be placed about 20 m from the first barrier wall.

6. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} Figure 2 shows the entrance to a road tunnel. The maximum height of the tunnel is measured as 5 metres and the width of the base of the tunnel is measured as 6 metres. Figure 3 shows a quadratic curve \(B C A\) used to model this entrance.
The points \(A , O , B\) and \(C\) are assumed to lie in the same vertical plane and the ground \(A O B\) is assumed to be horizontal.

1. Find an equation for curve \(B C A\). A coach has height 4.1 m and width 2.4 m .
2. Determine whether or not it is possible for the coach to enter the tunnel.
3. Suggest a reason why this model may not be suitable to determine whether or not the coach can pass through the tunnel.

\includegraphics{figure_5} \includegraphics{figure_6} A suspension bridge cable \(PQR\) hangs between the tops of two vertical towers, \(AP\) and \(BR\), as shown in Figure 5. A walkway \(AOB\) runs between the bases of the towers, directly under the cable. The towers are 100 m apart and each tower is 24 m high. At the point \(O\), midway between the towers, the cable is 4 m above the walkway. The points \(P\), \(Q\), \(R\), \(A\), \(O\) and \(B\) are assumed to lie in the same vertical plane and \(AOB\) is assumed to be horizontal. Figure 6 shows a symmetric quadratic curve \(PQR\) used to model this cable. Given that \(O\) is the origin,

1. find an equation for the curve \(PQR\). [3] Lee can safely inspect the cable up to a height of 12 m above the walkway. A defect is reported on the cable at a location 19 m horizontally from one of the towers.
2. Determine whether, according to the model, Lee can safely inspect this defect. [2]