Show formula then optimise: cylinder/prism (single variable)

16. \begin{figure}[h] \end{figure} Figure 4 shows the design for a container in the shape of a hollow triangular prism. The container is open at the top, which is labelled \(A B C D\). The sides of the container, \(A B F E\) and \(D C F E\), are rectangles. The ends of the container, \(A D E\) and \(B C F\), are congruent right-angled triangles, as shown in Figure 4. The ends of the container are vertical and the edge \(E F\) is horizontal. The edges \(A E , D E\) and \(E F\) have lengths \(4 x\) metres, \(3 x\) metres and \(l\) metres respectively. Given that the container has a capacity of \(0.75 \mathrm {~m} ^ { 3 }\) and is made of material of negligible thickness,

1. show that the internal surface area of the container, \(S \mathrm {~m} ^ { 2 }\), is given by $$S = 12 x ^ { 2 } + \frac { 7 } { 8 x }$$
2. Use calculus to find the value of \(x\), for which \(S\) is a minimum. Give your answer to 3 significant figures.
3. Justify that the value of \(x\) found in part (b) gives a minimum value for \(S\). Using the value of \(x\) found in part (b), find to 2 decimal places,
4. 1. the length of the edge \(A D\),
   2. the length of the edge \(C D\).

      END

14. \begin{figure}[h] \end{figure} Figure 6 shows a solid triangular prism \(A B C D E F\) in which \(A B = 2 x \mathrm {~cm}\) and \(C D = l \mathrm {~cm}\). The cross section \(A B C\) is an equilateral triangle. The rectangle \(B C D F\) is horizontal and the triangles \(A B C\) and \(D E F\) are vertical.
The total surface area of the prism is \(S \mathrm {~cm} ^ { 2 }\) and the volume of the prism is \(V \mathrm {~cm} ^ { 3 }\).

1. Show that \(S = 2 x ^ { 2 } \sqrt { 3 } + 6 x l\) Given that \(S = 960\),
2. show that \(V = 160 x \sqrt { 3 } - x ^ { 3 }\)
3. Use calculus to find the maximum value of \(V\), giving your answer to the nearest integer.
4. Justify that the value of \(V\) found in part (c) is a maximum. \includegraphics[max width=\textwidth, alt={}, center]{b85872d4-00b2-499b-9765-f7559d3de66a-24_63_52_2690_1886}

15. \begin{figure}[h] \end{figure} Figure 5 shows a design for a water barrel.
It is in the shape of a right circular cylinder with height \(h \mathrm {~cm}\) and radius \(r \mathrm {~cm}\). The barrel has a base but has no lid, is open at the top and is made of material of negligible thickness. The barrel is designed to hold \(60000 \mathrm {~cm} ^ { 3 }\) of water when full.

1. Show that the total external surface area, \(S \mathrm {~cm} ^ { 2 }\), of the barrel is given by the formula $$S = \pi r ^ { 2 } + \frac { 120000 } { r }$$
2. Use calculus to find the minimum value of \(S\), giving your answer to 3 significant figures.
3. Justify that the value of \(S\) you found in part (b) is a minimum.

15. \begin{figure}[h] \end{figure} Figure 4 shows a solid wooden block. The block is a right prism with length \(h \mathrm {~cm}\). The cross-section of the block is a semi-circle with radius \(r \mathrm {~cm}\). The total surface area of the block, including the curved surface, the two semi-circular ends and the rectangular base, is \(200 \mathrm {~cm} ^ { 2 }\)

1. Show that the volume \(V \mathrm {~cm} ^ { 3 }\) of the block is given by $$V = \frac { \pi r \left( 200 - \pi r ^ { 2 } \right) } { 4 + 2 \pi }$$
2. Use calculus to find the maximum value of \(V\). Give your answer to the nearest \(\mathrm { cm } ^ { 3 }\).
3. Justify, by further differentiation, that the value of \(V\) that you have found is a maximum.

1. In this question you must show all stages of your working.

\section*{Solutions based entirely on calculator technology are not acceptable.} \begin{figure}[h] \end{figure} A brick is in the shape of a cuboid with width \(x \mathrm {~cm}\) ,length \(3 x \mathrm {~cm}\) and height \(h \mathrm {~cm}\) ,as shown in Figure 2. The volume of the brick is \(972 \mathrm {~cm} ^ { 3 }\)

1. Show that the surface area of the brick,\(S \mathrm {~cm} ^ { 2 }\) ,is given by $$S = 6 x ^ { 2 } + \frac { 2592 } { x }$$
2. Find \(\frac { \mathrm { d } S } { \mathrm {~d} x }\)
3. Hence find the value of \(x\) for which \(S\) is stationary.
4. Find \(\frac { \mathrm { d } ^ { 2 } S } { \mathrm {~d} x ^ { 2 } }\) and hence show that the value of \(x\) found in part(c)gives the minimum value of \(S\) .
5. Hence find the minimum surface area of the brick.

9. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of a square based, open top box.
The height of the box is \(h \mathrm {~cm}\), and the base edges each have length \(l \mathrm {~cm}\).
Given that the volume of the box is \(250000 \mathrm {~cm} ^ { 3 }\)

1. show that the external surface area, \(S \mathrm {~cm} ^ { 2 }\), of the box is given by $$S = \frac { 250000 } { h } + 2000 \sqrt { h }$$
2. Use algebraic differentiation to show that \(S\) has a stationary point when \(h = 250 ^ { k }\) where \(k\) is a rational constant to be found.
3. Justify by further differentiation that this value of \(h\) gives the minimum external surface area of the box.
   \includegraphics[max width=\textwidth, alt={}]{515f245f-9c5b-4263-ab2c-0a4f96f3bff0-32_2647_1838_118_116}

10. A solid right circular cylinder has radius \(r \mathrm {~cm}\) and height \(h \mathrm {~cm}\). The total surface area of the cylinder is \(800 \mathrm {~cm} ^ { 2 }\).

1. Show that the volume, \(V \mathrm {~cm} ^ { 3 }\), of the cylinder is given by $$V = 400 r - \pi r ^ { 3 }$$ Given that \(r\) varies,
2. use calculus to find the maximum value of \(V\), to the nearest \(\mathrm { cm } ^ { 3 }\).
3. Justify that the value of \(V\) you have found is a maximum. \includegraphics[max width=\textwidth, alt={}, center]{12e54724-64a3-4dc0-b7d5-6ef6cc04124c-16_103_63_2477_1873}

10. \begin{figure}[h] \end{figure} Figure 4 shows a solid brick in the shape of a cuboid measuring \(2 x \mathrm {~cm}\) by \(x \mathrm {~cm}\) by \(y \mathrm {~cm}\). The total surface area of the brick is \(600 \mathrm {~cm} ^ { 2 }\).

1. Show that the volume, \(V \mathrm {~cm} ^ { 3 }\), of the brick is given by $$V = 200 x - \frac { 4 x ^ { 3 } } { 3 }$$ Given that \(x\) can vary,
2. use calculus to find the maximum value of \(V\), giving your answer to the nearest \(\mathrm { cm } ^ { 3 }\).
3. Justify that the value of \(V\) you have found is a maximum.

9. \begin{figure}[h] \end{figure} Figure 2 shows a closed box used by a shop for packing pieces of cake. The box is a right prism of height \(h \mathrm {~cm}\). The cross section is a sector of a circle. The sector has radius \(r \mathrm {~cm}\) and angle 1 radian. The volume of the box is \(300 \mathrm {~cm} ^ { 3 }\).

1. Show that the surface area of the box, \(S \mathrm {~cm} ^ { 2 }\), is given by $$S = r ^ { 2 } + \frac { 1800 } { r }$$
2. Use calculus to find the value of \(r\) for which \(S\) is stationary.
3. Prove that this value of \(r\) gives a minimum value of \(S\).
4. Find, to the nearest \(\mathrm { cm } ^ { 2 }\), this minimum value of \(S\).

10. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} Figure 4 shows a closed letter box \(A B F E H G C D\), which is made to be attached to a wall of a house. The letter box is a right prism of length \(y \mathrm {~cm}\) as shown in Figure 4. The base \(A B F E\) of the prism is a rectangle. The total surface area of the six faces of the prism is \(S \mathrm {~cm} ^ { 2 }\). The cross section \(A B C D\) of the letter box is a trapezium with edges of lengths \(D A = 9 x \mathrm {~cm}\), \(A B = 4 x \mathrm {~cm} , B C = 6 x \mathrm {~cm}\) and \(C D = 5 x \mathrm {~cm}\) as shown in Figure 5.
The angle \(D A B = 90 ^ { \circ }\) and the angle \(A B C = 90 ^ { \circ }\). The volume of the letter box is \(9600 \mathrm {~cm} ^ { 3 }\).

1. Show that $$y = \frac { 320 } { x ^ { 2 } }$$
2. Hence show that the surface area of the letter box, \(S \mathrm {~cm} ^ { 2 }\), is given by $$S = 60 x ^ { 2 } + \frac { 7680 } { x }$$
3. Use calculus to find the minimum value of \(S\).
4. Justify, by further differentiation, that the value of \(S\) you have found is a minimum.

10. \includegraphics[max width=\textwidth, alt={}, center]{6ef55dbd-f18d-4264-b80c-d181473ca7b3-3_531_786_246_523} The diagram shows an open-topped cylindrical container made from cardboard. The cylinder is of height \(h \mathrm {~cm}\) and base radius \(r \mathrm {~cm}\). Given that the area of card used to make the container is \(192 \pi \mathrm {~cm} ^ { 2 }\),

1. show that the capacity of the container, \(\mathrm { V } \mathrm { cm } ^ { 3 }\), is given by $$V = 96 \pi r - \frac { 1 } { 2 } \pi r ^ { 3 } .$$
2. Find the value of \(r\) for which \(V\) is stationary.
3. Find the corresponding value of \(V\) in terms of \(\pi\).
4. Determine whether this is a maximum or a minimum value of \(V\).

3

1. The standard formulae for the volume \(V\) and total surface area \(A\) of a solid cylinder of radius \(r\) and height \(h\) are $$V = \pi r ^ { 2 } h \quad \text { and } \quad A = 2 \pi r ^ { 2 } + 2 \pi r h .$$ Use these to show that, for a cylinder with \(A = 200\), $$V = 100 r - \pi r ^ { 3 }$$
2. Find \(\frac { \mathrm { d } V } { \mathrm {~d} r }\) and \(\frac { \mathrm { d } ^ { 2 } V } { \mathrm {~d} r ^ { 2 } }\).
3. Use calculus to find the value of \(r\) that gives a maximum value for \(V\) and hence find this maximum value, giving your answers correct to 3 significant figures.

3 A cylindrical metal tin of radius \(r \mathrm {~cm}\) is closed at both ends. It has a volume of \(16000 \pi \mathrm {~cm} ^ { 3 }\).

1. Show that its total surface area, \(A \mathrm {~cm} ^ { 2 }\), is given by \(A = 2 \pi r ^ { 2 } + 32000 \pi r ^ { - 1 }\).
2. Use calculus to determine the minimum total surface area of the tin. You should justify that it is a minimum.

7 \includegraphics[max width=\textwidth, alt={}, center]{7fc02f90-8f8b-4153-bba1-dc0807124e96-5_421_944_251_242} The diagram shows a model for the roof of a toy building. The roof is in the form of a solid triangular prism \(A B C D E F\). The base \(A C F D\) of the roof is a horizontal rectangle, and the crosssection \(A B C\) of the roof is an isosceles triangle with \(A B = B C\). The lengths of \(A C\) and \(C F\) are \(2 x \mathrm {~cm}\) and \(y \mathrm {~cm}\) respectively, and the height of \(B E\) above the base of the roof is \(x \mathrm {~cm}\). The total surface area of the five faces of the roof is \(600 \mathrm {~cm} ^ { 2 }\) and the volume of the roof is \(V \mathrm {~cm} ^ { 3 }\).

1. Show that \(V = k x \left( 300 - x ^ { 2 } \right)\), where \(k = \sqrt { a } + b\) and \(a\) and \(b\) are integers to be determined.
2. Use differentiation to determine the value of \(x\) for which the volume of the roof is a maximum.
3. Find the maximum volume of the roof. Give your answer in \(\mathrm { cm } ^ { 3 }\), correct to the nearest integer.
4. Explain why, for this roof, \(x\) must be less than a certain value, which you should state.

6 The diagram shows a block of wood in the shape of a prism with triangular cross-section. The end faces are right-angled triangles with sides of lengths \(3 x \mathrm {~cm}\), \(4 x \mathrm {~cm}\) and \(5 x \mathrm {~cm}\), and the length of the prism is \(y \mathrm {~cm}\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{66813123-3876-4484-aad1-4bfc09bb1508-7_394_825_459_548} The total surface area of the five faces is \(144 \mathrm {~cm} ^ { 2 }\).

1. 1. Show that \(x y + x ^ { 2 } = 12\).
   2. Hence show that the volume of the block, \(V \mathrm {~cm} ^ { 3 }\), is given by $$V = 72 x - 6 x ^ { 3 }$$
2. 1. Find \(\frac { \mathrm { d } V } { \mathrm {~d} x }\).
   2. Show that \(V\) has a stationary value when \(x = 2\).
3. Find \(\frac { \mathrm { d } ^ { 2 } V } { \mathrm {~d} x ^ { 2 } }\) and hence determine whether \(V\) has a maximum value or a minimum value when \(x = 2\).
   (2 marks)

4 The diagram shows a solid cuboid with sides of lengths \(x \mathrm {~cm} , 3 x \mathrm {~cm}\) and \(y \mathrm {~cm}\). \includegraphics[max width=\textwidth, alt={}, center]{dbc25177-4a28-480f-93d5-41acb2a2d28c-3_349_472_376_769} The total surface area of the cuboid is \(32 \mathrm {~cm} ^ { 2 }\).

1. 1. Show that \(3 x ^ { 2 } + 4 x y = 16\).
   2. Hence show that the volume, \(V \mathrm {~cm} ^ { 3 }\), of the cuboid is given by $$V = 12 x - \frac { 9 x ^ { 3 } } { 4 }$$
2. Find \(\frac { \mathrm { d } V } { \mathrm {~d} x }\).
3. 1. Verify that a stationary value of \(V\) occurs when \(x = \frac { 4 } { 3 }\).
   2. Find \(\frac { \mathrm { d } ^ { 2 } V } { \mathrm {~d} x ^ { 2 } }\) and hence determine whether \(V\) has a maximum value or a minimum value when \(x = \frac { 4 } { 3 }\).

6. A container made from thin metal is in the shape of a right circular cylinder with height \(h \mathrm {~cm}\) and base radius \(r \mathrm {~cm}\). The container has no lid. When full of water, the container holds \(500 \mathrm {~cm} ^ { 3 }\) of water.

1. Show that the exterior surface area, \(A \mathrm {~cm} ^ { 2 }\), of the container is given by \(A = \pi r ^ { 2 } + \frac { 1000 } { r }\).
2. Find the value of \(r\) for which \(A\) is a minimum.
3. Prove that this value of \(r\) gives a minimum value of \(A\).
4. Calculate the minimum value of \(A\), giving your answer to the nearest integer.

A pencil holder is in the shape of an open circular cylinder of radius \(r\) cm and height \(h\) cm. The surface area of the cylinder (including the base) is 250 cm\(^2\).

1. Show that the volume, V cm\(^3\), of the cylinder is given by \(V = 125r - \frac{\pi r^3}{2}\). [4]
2. Use calculus to find the value of \(r\) for which \(V\) has a stationary value. [3]
3. Prove that the value of \(r\) you found in part (b) gives a maximum value for \(V\). [2]
4. Calculate, to the nearest cm\(^3\), the maximum volume of the pencil holder. [2]

A cylinder is to be cut out of the circular face of a solid hemisphere. The cylinder and the hemisphere have the same axis of symmetry. The cylinder has height \(h\) and the hemisphere has a radius of \(R\). \includegraphics{figure_9}

1. Show that the volume, \(V\), of the cylinder is given by $$V = \pi R^2 h - \pi h^3$$ [3 marks]
2. Find the maximum volume of the cylinder in terms of \(R\). Fully justify your answer. [7 marks]