Optimization with constraints

7. \begin{figure}[h] \end{figure} Figure 1 shows a rectangular sheet of metal of negligible thickness, which measures 25 cm by 15 cm . Squares of side \(x \mathrm {~cm}\) are cut from each corner of the sheet and the remainder is folded along the dotted lines to make an open cuboid box, as shown in Figure 2.

1. Show that the volume, \(V \mathrm {~cm} ^ { 3 }\), of the box is given by $$V = 4 x ^ { 3 } - 80 x ^ { 2 } + 375 x$$
2. Use calculus to find the value of \(x\), to 3 significant figures, for which the volume of the box is a maximum.
3. Justify that this value of \(x\) gives a maximum value for \(V\).
4. Find, to 3 significant figures, the maximum volume of the box.
   \section*{8.} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{6f9ace43-747b-419f-a9d1-d30165d77379-22_670_1004_292_392} \captionsetup{labelformat=empty} \caption{Figure 3}
   \end{figure} Figure 3 shows a sketch of the curve with equation \(y = \mathrm { f } ( x ) , x \in \mathbb { R }\). The curve crosses the \(y\)-axis at the point \(( 0,5 )\) and crosses the \(x\)-axis at the point \(( 6,0 )\). The curve has a minimum point at \(( 1,3 )\) and a maximum point at \(( 4,7 )\). On separate diagrams, sketch the curve with equation

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.

15. \begin{figure}[h] \end{figure} Figure 2 shows a plan for a garden.
The garden consists of two identical rectangles of width \(y \mathrm {~m}\) and length \(x \mathrm {~m}\), joined to a sector of a circle with radius \(x \mathrm {~m}\) and angle 0.8 radians, as shown in Figure 2. The area of the garden is \(60 \mathrm {~m} ^ { 2 }\).

1. Show that the perimeter, \(P \mathrm {~m}\), of the garden is given by $$P = 2 x + \frac { 120 } { x }$$
2. Use calculus to find the exact minimum value for \(P\), giving your answer in the form \(a \sqrt { b }\), where \(a\) and \(b\) are integers.
3. Justify that the value of \(P\) found in part (b) is the minimum. \includegraphics[max width=\textwidth, alt={}, center]{1f61f78b-5e77-4758-8ad5-ea00c7dfea2b-49_83_59_2636_1886}

13. \begin{figure}[h] \end{figure} Figure 3 shows a flowerbed. Its shape is a quarter of a circle of radius \(x\) metres with two equal rectangles attached to it along its radii. Each rectangle has length equal to \(x\) metres and width equal to \(y\) metres. Given that the area of the flowerbed is \(4 \mathrm {~m} ^ { 2 }\),

1. show that $$y = \frac { 16 - \pi x ^ { 2 } } { 8 x }$$
2. Hence show that the perimeter \(P\) metres of the flowerbed is given by the equation $$P = \frac { 8 } { x } + 2 x$$
3. Use calculus to find the minimum value of \(P\).

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}

1. A diesel lorry is driven from Birmingham to Bury at a steady speed of v kilometres per hour. The total cost of the journey, \(\pounds C\), is given by

$$C = \frac { 1400 } { v } + \frac { 2 v } { 7 } .$$

1. Find the value of \(v\) for which \(C\) is a minimum.
2. Find \(\frac { \mathrm { d } ^ { 2 } C } { \mathrm {~d} v ^ { 2 } }\) and hence verify that \(C\) is a minimum for this value of \(v\).
3. Calculate the minimum total cost of the journey.

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}

1. The volume \(V \mathrm {~cm} ^ { 3 }\) of a box, of height \(x \mathrm {~cm}\), is given by

$$V = 4 x ( 5 - x ) ^ { 2 } , \quad 0 < x < 5$$

1. Find \(\frac { \mathrm { d } V } { \mathrm {~d} x }\).
2. Hence find the maximum volume of the box.
3. Use calculus to justify that the volume that you found in part (b) is a maximum.

8. \begin{figure}[h] \end{figure} Figure 3 shows a flowerbed. Its shape is a quarter of a circle of radius \(x\) metres with two equal rectangles attached to it along its radii. Each rectangle has length equal to \(x\) metres and width equal to \(y\) metres. Given that the area of the flowerbed is \(4 \mathrm {~m} ^ { 2 }\),

1. show that $$y = \frac { 16 - \pi x ^ { 2 } } { 8 x }$$
2. Hence show that the perimeter \(P\) metres of the flowerbed is given by the equation $$P = \frac { 8 } { x } + 2 x$$
3. Use calculus to find the minimum value of \(P\).
4. Find the width of each rectangle when the perimeter is a minimum. Give your answer to the nearest centimetre.

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 \(S\).

9. \begin{figure}[h] \end{figure} Figure 4 shows the plan of a pool. The shape of the pool \(A B C D E F A\) consists of a rectangle \(B C E F\) joined to an equilateral triangle \(B F A\) and a semi-circle \(C D E\), as shown in Figure 4. Given that \(A B = x\) metres, \(E F = y\) metres, and the area of the pool is \(50 \mathrm {~m} ^ { 2 }\),

1. show that $$y = \frac { 50 } { x } - \frac { x } { 8 } ( \pi + 2 \sqrt { } 3 )$$
2. Hence show that the perimeter, \(P\) metres, of the pool is given by $$P = \frac { 100 } { x } + \frac { x } { 4 } ( \pi + 8 - 2 \sqrt { } 3 )$$
3. Use calculus to find the minimum value of \(P\), giving your answer to 3 significant figures.
4. Justify, by further differentiation, that the value of \(P\) that you have found is a minimum.

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.

9. A solid glass cylinder, which is used in an expensive laser amplifier, has a volume of \(75 \pi \mathrm {~cm} ^ { 3 }\).
The cost of polishing the surface area of this glass cylinder is \(\pounds 2\) per \(\mathrm { cm } ^ { 2 }\) for the curved surface area and \(\pounds 3\) per \(\mathrm { cm } ^ { 2 }\) for the circular top and base areas. Given that the radius of the cylinder is \(r \mathrm {~cm}\),

1. show that the cost of the polishing, \(\pounds C\), is given by $$C = 6 \pi r ^ { 2 } + \frac { 300 \pi } { r }$$
2. Use calculus to find the minimum cost of the polishing, giving your answer to the nearest pound.
3. Justify that the answer that you have obtained in part (b) is a minimum.

9. \begin{figure}[h] \end{figure} Figure 4 shows a plan view of a sheep enclosure.
The enclosure \(A B C D E A\), as shown in Figure 4, consists of a rectangle \(B C D E\) joined to an equilateral triangle \(B F A\) and a sector \(F E A\) of a circle with radius \(x\) metres and centre \(F\). The points \(B , F\) and \(E\) lie on a straight line with \(F E = x\) metres and \(10 \leqslant x \leqslant 25\)

1. Find, in \(\mathrm { m } ^ { 2 }\), the exact area of the sector \(F E A\), giving your answer in terms of \(x\), in its simplest form. Given that \(B C = y\) metres, where \(y > 0\), and the area of the enclosure is \(1000 \mathrm {~m} ^ { 2 }\),
2. show that $$y = \frac { 500 } { x } - \frac { x } { 24 } ( 4 \pi + 3 \sqrt { 3 } )$$
3. Hence show that the perimeter \(P\) metres of the enclosure is given by $$P = \frac { 1000 } { x } + \frac { x } { 12 } ( 4 \pi + 36 - 3 \sqrt { 3 } )$$
4. Use calculus to find the minimum value of \(P\), giving your answer to the nearest metre.
5. Justify, by further differentiation, that the value of \(P\) you have found is a minimum.

9. Figure 3 $$( x + 1 ) ^ { 2 }$$ Figure 3 shows a triangle \(P Q R\). The size of angle \(Q P R\) is \(30 ^ { \circ }\), the length of \(P Q\) is \(( x + 1 )\) and the length of \(P R\) is \(( 4 - x ) ^ { 2 }\), where \(X \in \Re\).

1. Show that the area \(A\) of the triangle is given by \(A = \frac { 1 } { 4 } \left( x ^ { 3 } - 7 x ^ { 2 } + 8 x + 16 \right)\)
2. Use calculus to prove that the area of \(\triangle P Q R\) is a maximum when \(x = \frac { 2 } { 3 }\). Explain clearly how you know that this value of \(x\) gives the maximum area.
3. Find the maximum area of \(\triangle P Q R\).
4. Find the length of \(Q R\) when the area of \(\triangle P Q R\) is a maximum. END

8 A cuboid has a volume of \(8 \mathrm {~m} ^ { 3 }\). The base of the cuboid is square with sides of length \(x\) metres. The surface area of the cuboid is \(A \mathrm {~m} ^ { 2 }\).

1. Show that \(A = 2 x ^ { 2 } + \frac { 32 } { x }\).
2. Find \(\frac { \mathrm { d } A } { \mathrm {~d} x }\).
3. Find the value of \(x\) which gives the smallest surface area of the cuboid, justifying your answer.

5
\includegraphics[max width=\textwidth, alt={}, center]{581ef815-59f0-434e-a7ec-9128e74c0323-2_256_1113_1366_516} The diagram shows a rectangular enclosure, with a wall forming one side. A rope, of length 20 metres, is used to form the remaining three sides. The width of the enclosure is x metres.

1. Show that the enclosed area, \(\mathrm { Am } ^ { 2 }\), is given by $$A = 20 x - 2 x ^ { 2 } .$$
2. Use differentiation to find the maximum value of A .

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 \(V\).

12 Fig. 12 shows a window. The base and sides are parts of a rectangle with dimensions \(2 x\) metres horizontally by \(y\) metres vertically. The top is a semicircle of radius \(x\) metres. The perimeter of the window is 10 metres. \begin{figure}[h] \end{figure}

1. Express \(y\) as a function of \(x\).
2. Find the total area, \(A \mathrm {~m} ^ { 2 }\), in terms of \(x\) and \(y\). Use your answer to part (i) to show that this simplifies to $$A = 10 x - 2 x ^ { 2 } - \frac { 1 } { 2 } \pi x ^ { 2 }$$
3. Prove that for the maximum value of \(A\), \(y = x\) exactly.
   \section*{MEI STRUCTURED MATHEMATICS } \section*{CONCEPTS FOR ADVANCED MATHEMATICS, C2} \section*{Practice Paper C2-B
   Insert sheet for question 11} 11 Speed-time graph with the first two points plotted.
   \includegraphics[max width=\textwidth, alt={}, center]{73d1c02b-1b7b-426d-a171-c762597cfed4-5_768_1772_1389_205}