Applied context: real-world solid

4. \begin{figure}[h] \end{figure} The curve \(C\) with equation \(y = \frac { 2 } { ( 4 + 3 x ) } , x > - \frac { 4 } { 3 }\) is shown in Figure 1
The region bounded by the curve, the \(x\)-axis and the lines \(x = - 1\) and \(x = \frac { 2 } { 3 }\), is shown shaded in Figure 1 This region is rotated through 360 degrees about the \(x\)-axis.

1. Use calculus to find the exact value of the volume of the solid generated. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{101ec3c2-699e-4c74-bfdc-d8c610646571-05_583_433_1398_753} \captionsetup{labelformat=empty} \caption{Figure 2}
   \end{figure} Figure 2 shows a candle with axis of symmetry \(A B\) where \(A B = 15 \mathrm {~cm}\). \(A\) is a point at the centre of the top surface of the candle and \(B\) is a point at the centre of the base of the candle. The candle is geometrically similar to the solid generated in part (a).
2. Find the volume of this candle.

7. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} Figure 3 shows the design of a doorknob.
The shape of the doorknob is formed by rotating the curve shown in Figure 4 through \(360 ^ { \circ }\) about the \(x\)-axis, where the units are centimetres. The equation of the curve is given by $$\mathrm { f } ( x ) = \frac { 1 } { 4 } ( 4 - x ) \mathrm { e } ^ { x } \quad 0 \leqslant x \leqslant 4$$

1. Show that the volume, \(V \mathrm {~cm} ^ { 3 }\), of the doorknob is given by $$V = K \int _ { 0 } ^ { 4 } \left( x ^ { 2 } - 8 x + 16 \right) \mathrm { e } ^ { 2 x } \mathrm {~d} x$$ where \(K\) is a constant to be found.
2. Hence, find the exact value of the volume of the doorknob. Give your answer in the form \(p \pi \left( \mathrm { e } ^ { q } + r \right) \mathrm { cm } ^ { 3 }\) where \(p , q\) and \(r\) are simplified rational numbers to be found.

2. \begin{figure}[h] \end{figure} The curve with equation \(y = \frac { 1 } { 3 ( 1 + 2 x ) } , x > - \frac { 1 } { 2 }\), is shown in Figure 1.
The region bounded by the lines \(x = - \frac { 1 } { 4 } , x = \frac { 1 } { 2 }\), the \(x\)-axis and the curve is shown shaded in Figure 1. This region is rotated through 360 degrees about the \(x\)-axis.

1. Use calculus to find the exact value of the volume of the solid generated. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{d366e541-15f6-4fb5-9afb-faf6120f1a1c-03_383_447_1411_753}
   \end{figure} Figure 2 shows a paperweight with axis of symmetry \(A B\) where \(A B = 3 \mathrm {~cm}\). \(A\) is a point on the top surface of the paperweight, and \(B\) is a point on the base of the paperweight. The paperweight is geometrically similar to the solid in part (a).
2. Find the volume of this paperweight.

9

1. A tunnel is 100 m long. Its cross-section, shown in Fig. 9.1, is modelled by the curve $$y = \frac { 1 } { 4 } \left( 10 x - x ^ { 2 } \right) ,$$ where \(x\) and \(y\) are horizontal and vertical distances in metres. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{1e43ddbe-ae95-467b-a527-351ab8a4c4fe-004_506_812_676_653} \captionsetup{labelformat=empty} \caption{Figure 9.1}
   \end{figure} Using this model,
   (A) find the greatest height of the tunnel,
   (B) explain why \(100 \int _ { 0 } ^ { 10 } y \mathrm {~d} x\) gives the volume, in cubic metres, of earth removed to make the tunnel. Calculate this volume.
2. The roof of the tunnel is re-shaped to allow for larger vehicles. Fig. 9.2 shows the new crosssection. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{1e43ddbe-ae95-467b-a527-351ab8a4c4fe-004_513_1256_1894_575} \captionsetup{labelformat=empty} \caption{Fig. 9.2}
   \end{figure} Use the trapezium rule with 5 strips to estimate the new cross-sectional area.
   Hence estimate the volume of earth removed when the tunnel is re-shaped.

9

1. A tunnel is 100 m long. Its cross-section, shown in Fig. 9.1, is modelled by the curve $$y = \frac { 1 } { 4 } \left( 10 x - x ^ { 2 } \right) ,$$ where \(x\) and \(y\) are horizontal and vertical distances in metres. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{72b4624f-e716-4a37-96f3-01b46e0bd0fd-5_506_812_676_653} \captionsetup{labelformat=empty} \caption{Figure 9.1}
   \end{figure} Using this model,
   (A) find the greatest height of the tunnel,
   (B) explain why \(100 \int _ { 0 } ^ { 10 } y \mathrm {~d} x\) gives the volume, in cubic metres, of earth removed to make the tunnel. Calculate this volume.
2. The roof of the tunnel is re-shaped to allow for larger vehicles. Fig. 9.2 shows the new crosssection. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{72b4624f-e716-4a37-96f3-01b46e0bd0fd-5_513_1256_1894_575} \captionsetup{labelformat=empty} \caption{Fig. 9.2}
   \end{figure} Use the trapezium rule with 5 strips to estimate the new cross-sectional area.
   Hence estimate the volume of earth removed when the tunnel is re-shaped.

9 The shape of a vase can be modelled by rotating the curve with equation \(16 x ^ { 2 } - ( y - 8 ) ^ { 2 } = 32\) between \(y = 0\) and \(y = 16\) completely about the \(\boldsymbol { y }\)-axis. \includegraphics[max width=\textwidth, alt={}, center]{063bbfa5-df49-44a1-8143-5e076397f63f-09_890_1210_1555_424} The vase has a base.
Find the volume of water needed to fill the vase, giving your answer as an exact value.

9. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} A mathematics student is modelling the profile of a glass bottle of water. Figure 1 shows a sketch of a central vertical cross-section \(A B C D E F G H A\) of the bottle with the measurements taken by the student. The horizontal cross-section between \(C F\) and \(D E\) is a circle of diameter 8 cm and the horizontal cross-section between \(B G\) and \(A H\) is a circle of diameter 2 cm . The student thinks that the curve \(G F\) could be modelled as a curve with equation $$y = a x ^ { 2 } + b \quad 1 \leqslant x \leqslant 4$$ where \(a\) and \(b\) are constants and \(O\) is the fixed origin, as shown in Figure 2.

1. Find the value of \(a\) and the value of \(b\) according to the model.
2. Use the model to find the volume of water that the bottle can contain.
3. State a limitation of the model. The label on the bottle states that the bottle holds approximately \(750 \mathrm {~cm} ^ { 3 }\) of water.
4. Use this information and your answer to part (b) to evaluate the model, explaining your reasoning.

9. \begin{figure}[h] \end{figure} Figure 2 shows the vertical cross-section, \(A O B C D E\), through the centre of a wax candle.
In a model, the candle is formed by rotating the region bounded by the \(y\)-axis, the line \(O B\), the curve \(B C\), and the curve \(C D\) through \(360 ^ { \circ }\) about the \(y\)-axis. The point \(B\) has coordinates \(( 3,0 )\) and the point \(C\) has coordinates \(( 5,15 )\).
The units are in centimetres.
The curve \(B C\) is represented by the equation $$y = \frac { \sqrt { 225 x ^ { 2 } - 2025 } } { a } \quad 3 \leqslant x < 5$$ where \(a\) is a constant.

1. Determine the value of \(a\) according to this model. The curve \(C D\) is represented by the equation $$y = 16 - 0.04 x ^ { 2 } \quad 0 \leqslant x < 5$$
2. Using algebraic integration, determine, according to the model, the exact volume of wax that would be required to make the candle.
3. State a limitation of the model. When the candle was manufactured, \(700 \mathrm {~cm} ^ { 3 }\) of wax were required.
4. Use this information and your answer to part (b) to evaluate the model, explaining your reasoning.

8. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} Figure 1 shows a sketch of a 16 cm tall vase which has a flat circular base with diameter 8 cm and a circular opening of diameter 8 cm at the top. A student measures the circular cross-section halfway up the vase to be 8 cm in diameter.
The student models the shape of the vase by rotating a curve, shown in Figure 2, through \(360 ^ { \circ }\) about the \(x\)-axis.

1. State the value of \(a\) that should be used when setting up the model. Two possible equations are suggested for the curve in the model. $$\begin{array} { l l } \text { Model A } & y = a - 2 \sin \left( \frac { 45 } { 2 } x \right) ^ { \circ } \\ \text { Model B } & y = a + \frac { x ( x - 8 ) ( x + 8 ) } { 100 } \end{array}$$ For each model,
2. 1. find the distance from the base at which the widest part of the vase occurs,
   2. find the diameter of the vase at this widest point. The widest part of the vase has diameter 12 cm and is just over 3 cm from the base.
3. Using this information and making your reasoning clear, suggest which model is more appropriate.
4. Using algebraic integration, find the volume for the vase predicted by Model B. You must make your method clear. The student pours water from a full one litre jug into the vase and finds that there is 100 ml left in the jug when the vase is full.
5. Comment on the suitability of Model B in light of this information.

7. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} Figure 1 shows the central cross-section \(A O B C D\) of a circular bird bath, which is made of concrete. Measurements of the height and diameter of the bird bath, and the depth of the bowl of the bird bath have been taken in order to estimate the amount of concrete that was required to make this bird bath. Using these measurements, the cross-sectional curve CD, shown in Figure 2, is modelled as a curve with equation $$y = 1 + k x ^ { 2 } \quad - 0.2 \leqslant x \leqslant 0.2$$ where \(k\) is a constant and where \(O\) is the fixed origin.
The height of the bird bath measured 1.16 m and the diameter, \(A B\), of the base of the bird bath measured 0.40 m , as shown in Figure 1.

1. Suggest the maximum depth of the bird bath.
2. Find the value of \(k\).
3. Hence find the volume of concrete that was required to make the bird bath according to this model. Give your answer, in \(\mathrm { m } ^ { 3 }\), correct to 3 significant figures.
4. State a limitation of the model. It was later discovered that the volume of concrete used to make the bird bath was \(0.127 \mathrm {~m} ^ { 3 }\) correct to 3 significant figures.
5. Using this information and the answer to part (c), evaluate the model, explaining your reasoning.

1 In this question you must show detailed reasoning.

1. Show that \(\cosh ( 2 \ln 3 ) = \frac { 41 } { 9 }\). The region \(R\) is bounded by the curve with equation \(\mathrm { y } = \sqrt { \operatorname { sinhx } }\), the \(x\)-axis and the line with equation \(x = 2 \ln 3\) (see diagram). The units of the axes are centimetres. \includegraphics[max width=\textwidth, alt={}, center]{23e58e5e-bbaa-4932-aad0-89b3de6647b2-2_652_668_740_242} A manufacturer produces bell-shaped chocolate pieces. Each piece is modelled as being the shape of the solid formed by rotating \(R\) completely about the \(x\)-axis.
2. Determine, according to the model, the exact volume of one chocolate piece.

4 In this question you must show detailed reasoning. The region \(R\) is bounded by the curve with equation \(\mathrm { y } = \frac { 1 } { \sqrt { 3 \mathrm { x } ^ { 2 } - 3 \mathrm { x } + 1 } }\), the \(x\)-axis and the lines with equations \(x = \frac { 1 } { 2 }\) and \(x = 1\) (see diagram). The units of the axes are cm . \includegraphics[max width=\textwidth, alt={}, center]{7b2bfb4e-524f-4d1c-ae98-075c7fb404f9-3_778_1241_497_242} A pendant is to be made out of a precious metal. The shape of the pendant is modelled as the shape formed when \(R\) is rotated by \(2 \pi\) radians about the \(x\)-axis. Find the exact value of the volume of precious metal required to make the pendant, according to the model.

The curve \(y = 1 + x^3\) is denoted by \(C\).

1. A bowl is designed by rotating the arc of \(C\) joining the points \((0,1)\) and \((2,9)\) through four right angles about the \(y\)-axis. Calculate the capacity of the bowl. [5]
2. Another bowl with capacity 25 is to be designed by rotating the arc of \(C\) joining the points with \(y\) coordinates 1 and \(a\) through four right angles about the \(y\)-axis. Calculate the value of \(a\). [5]

A candlestick has base diameter \(8\) cm and height \(28\) cm, as shown in Figure \(9\). A model of the candlestick is shown in Figure \(10\), together with the equations that were used to create the model. \includegraphics{figure_7}

1. Show that the volume generated by rotating the shaded region (in Figure \(10\)) \(2\pi\) radians about the \(y\)-axis is \(\frac{112}{15}\pi\). [4]
2. Hence find the volume of metal needed to create the candlestick. [4]

\includegraphics{figure_9} A mathematics student is modelling the profile of a glass bottle of water. Figure 1 shows a sketch of a central vertical cross-section \(ABCDEFGHA\) of the bottle with measurements taken by the student. The horizontal cross-section between \(CF\) and \(DE\) is a circle of diameter 8 cm and the horizontal cross-section between \(BG\) and \(AH\) is a circle of diameter 2 cm. The student thinks that the curve \(GF\) could be modelled as a curve with equation $$y = ax^2 + b \qquad 1 \leq x \leq 4$$ where \(a\) and \(b\) are constants and \(O\) is the fixed origin, as shown in Figure 2.

1. Find the value of \(a\) and the value of \(b\) according to the model. [2]
2. Use the model to find the volume of water that the bottle can contain. [7]