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.

11 A fire crew is tackling a grass fire on horizontal ground. The crew directs a single jet of water which flows continuously from point \(A\).
\includegraphics[max width=\textwidth, alt={}, center]{7cca79eb-fd09-4ec0-8a1d-a7a38ca73f7a-14_505_967_450_539} The path of the jet can be modelled by the equation $$y = - 0.0125 x ^ { 2 } + 0.5 x - 2.55$$ where \(x\) metres is the horizontal distance of the jet from the fire truck at \(O\) and \(y\) metres is the height of the jet above the ground. The coordinates of point \(A\) are ( \(a , 0\) )
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1. 1. Find the value of \(a\).
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2. (ii) Find the horizontal distance from \(\boldsymbol { A }\) to the point where the jet hits the ground.
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3. Calculate the maximum vertical height reached by the jet.
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4. A vertical wall is located 11 metres horizontally from \(A\) in the direction of the jet. The height of the wall is 2.3 metres. Using the model, determine whether the jet passes over the wall, stating any necessary modelling assumption.