Optimization with constraint

8 A hollow circular cylinder, open at one end, is constructed of thin sheet metal. The total external surface area of the cylinder is \(192 \pi \mathrm {~cm} ^ { 2 }\). The cylinder has a radius of \(r \mathrm {~cm}\) and a height of \(h \mathrm {~cm}\).

1. Express \(h\) in terms of \(r\) and show that the volume, \(V \mathrm {~cm} ^ { 3 }\), of the cylinder is given by $$V = \frac { 1 } { 2 } \pi \left( 192 r - r ^ { 3 } \right) .$$ Given that \(r\) can vary,
2. find the value of \(r\) for which \(V\) has a stationary value,
3. find this stationary value and determine whether it is a maximum or a minimum.

8
\includegraphics[max width=\textwidth, alt={}, center]{22a31966-4433-4d7d-8a75-bcd536acfa24-4_543_511_264_817} The diagram shows a glass window consisting of a rectangle of height \(h \mathrm {~m}\) and width \(2 r \mathrm {~m}\) and a semicircle of radius \(r \mathrm {~m}\). The perimeter of the window is 8 m .

1. Express \(h\) in terms of \(r\).
2. Show that the area of the window, \(A \mathrm {~m} ^ { 2 }\), is given by $$A = 8 r - 2 r ^ { 2 } - \frac { 1 } { 2 } \pi r ^ { 2 } .$$ Given that \(r\) can vary,
3. find the value of \(r\) for which \(A\) has a stationary value,
4. determine whether this stationary value is a maximum or a minimum.

8 A solid rectangular block has a square base of side \(x \mathrm {~cm}\). The height of the block is \(h \mathrm {~cm}\) and the total surface area of the block is \(96 \mathrm {~cm} ^ { 2 }\).

1. Express \(h\) in terms of \(x\) and show that the volume, \(V \mathrm {~cm} ^ { 3 }\), of the block is given by $$V = 24 x - \frac { 1 } { 2 } x ^ { 3 }$$ Given that \(x\) can vary,
2. find the stationary value of \(V\),
3. determine whether this stationary value is a maximum or a minimum.

6 The variables \(x , y\) and \(z\) can take only positive values and are such that $$z = 3 x + 2 y \quad \text { and } \quad x y = 600 .$$

1. Show that \(z = 3 x + \frac { 1200 } { x }\).
2. Find the stationary value of \(z\) and determine its nature.

8 The volume of a solid circular cylinder of radius \(r \mathrm {~cm}\) is \(250 \pi \mathrm {~cm} ^ { 3 }\).

1. Show that the total surface area, \(S \mathrm {~cm} ^ { 2 }\), of the cylinder is given by $$S = 2 \pi r ^ { 2 } + \frac { 500 \pi } { r }$$
2. Given that \(r\) can vary, find the stationary value of \(S\).
3. Determine the nature of this stationary value.

9 The base of a cuboid has sides of length \(x \mathrm {~cm}\) and \(3 x \mathrm {~cm}\). The volume of the cuboid is \(288 \mathrm {~cm} ^ { 3 }\).

1. Show that the total surface area of the cuboid, \(A \mathrm {~cm} ^ { 2 }\), is given by $$A = 6 x ^ { 2 } + \frac { 768 } { x }$$
2. Given that \(x\) can vary, find the stationary value of \(A\) and determine its nature.

4 Variables \(u , x\) and \(y\) are such that \(u = 2 x ( y - x )\) and \(x + 3 y = 12\). Express \(u\) in terms of \(x\) and hence find the stationary value of \(u\).

1. Prove the identity \(\frac { \sin \theta - \cos \theta } { \sin \theta + \cos \theta } \equiv \frac { \tan \theta - 1 } { \tan \theta + 1 }\).
2. Hence solve the equation \(\frac { \sin \theta - \cos \theta } { \sin \theta + \cos \theta } = \frac { \tan \theta } { 6 }\), for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).

6 The horizontal base of a solid prism is an equilateral triangle of side \(x \mathrm {~cm}\). The sides of the prism are vertical. The height of the prism is \(h \mathrm {~cm}\) and the volume of the prism is \(2000 \mathrm {~cm} ^ { 3 }\).

1. Express \(h\) in terms of \(x\) and show that the total surface area of the prism, \(A \mathrm {~cm} ^ { 2 }\), is given by $$A = \frac { \sqrt { } 3 } { 2 } x ^ { 2 } + \frac { 24000 } { \sqrt { } 3 } x ^ { - 1 }$$
2. Given that \(x\) can vary, find the value of \(x\) for which \(A\) has a stationary value.
3. Determine, showing all necessary working, the nature of this stationary value.

6 A vacuum flask (for keeping drinks hot) is modelled as a closed cylinder in which the internal radius is \(r \mathrm {~cm}\) and the internal height is \(h \mathrm {~cm}\). The volume of the flask is \(1000 \mathrm {~cm} ^ { 3 }\). A flask is most efficient when the total internal surface area, \(A \mathrm {~cm} ^ { 2 }\), is a minimum.

1. Show that \(A = 2 \pi r ^ { 2 } + \frac { 2000 } { r }\).
2. Given that \(r\) can vary, find the value of \(r\), correct to 1 decimal place, for which \(A\) has a stationary value and verify that the flask is most efficient when \(r\) takes this value.

8 A solid rectangular block has a base which measures \(2 x \mathrm {~cm}\) by \(x \mathrm {~cm}\). The height of the block is \(y \mathrm {~cm}\) and the volume of the block is \(72 \mathrm {~cm} ^ { 3 }\).

1. Express \(y\) in terms of \(x\) and show that the total surface area, \(A \mathrm {~cm} ^ { 2 }\), of the block is given by $$A = 4 x ^ { 2 } + \frac { 216 } { x }$$ Given that \(x\) can vary,
2. find the value of \(x\) for which \(A\) has a stationary value,
3. find this stationary value and determine whether it is a maximum or a minimum.

9
\includegraphics[max width=\textwidth, alt={}, center]{dd2cb0ec-5df9-4d99-9e15-5ae1f1c07b96-4_387_903_799_623} The diagram shows an open container constructed out of \(200 \mathrm {~cm} ^ { 2 }\) of cardboard. The two vertical end pieces are isosceles triangles with sides \(5 x \mathrm {~cm} , 5 x \mathrm {~cm}\) and \(8 x \mathrm {~cm}\), and the two side pieces are rectangles of length \(y \mathrm {~cm}\) and width \(5 x \mathrm {~cm}\), as shown. The open top is a horizontal rectangle.

1. Show that \(y = \frac { 200 - 24 x ^ { 2 } } { 10 x }\).
2. Show that the volume, \(V \mathrm {~cm} ^ { 3 }\), of the container is given by \(V = 240 x - 28.8 x ^ { 3 }\). Given that \(x\) can vary,
3. find the value of \(x\) for which \(V\) has a stationary value,
4. determine whether it is a maximum or a minimum stationary value.

7
\includegraphics[max width=\textwidth, alt={}, center]{08729aab-586b-4210-94c9-77b1f6b1d873-3_385_360_1379_561}
\includegraphics[max width=\textwidth, alt={}, center]{08729aab-586b-4210-94c9-77b1f6b1d873-3_364_369_1379_1219} A wire, 80 cm long, is cut into two pieces. One piece is bent to form a square of side \(x \mathrm {~cm}\) and the other piece is bent to form a circle of radius \(r \mathrm {~cm}\) (see diagram). The total area of the square and the circle is \(A \mathrm {~cm} ^ { 2 }\).

1. Show that \(A = \frac { ( \pi + 4 ) x ^ { 2 } - 160 x + 1600 } { \pi }\).
2. Given that \(x\) and \(r\) can vary, find the value of \(x\) for which \(A\) has a stationary value.

7
\includegraphics[max width=\textwidth, alt={}, center]{b566719c-216e-41e5-8431-da77e1dad73e-3_301_485_264_829} A piece of wire of length 50 cm is bent to form the perimeter of a sector \(P O Q\) of a circle. The radius of the circle is \(r \mathrm {~cm}\) and the angle \(P O Q\) is \(\theta\) radians (see diagram).

1. Express \(\theta\) in terms of \(r\) and show that the area, \(A \mathrm {~cm} ^ { 2 }\), of the sector is given by $$A = 25 r - r ^ { 2 } .$$
2. Given that \(r\) can vary, find the stationary value of \(A\) and determine its nature.

8
\includegraphics[max width=\textwidth, alt={}, center]{73c0c113-8f35-4e7f-ad5d-604602498b71-3_314_803_751_671} The diagram shows a metal plate consisting of a rectangle with sides \(x \mathrm {~cm}\) and \(y \mathrm {~cm}\) and a quarter-circle of radius \(x \mathrm {~cm}\). The perimeter of the plate is 60 cm .

1. Express \(y\) in terms of \(x\).
2. Show that the area of the plate, \(A \mathrm {~cm} ^ { 2 }\), is given by \(A = 30 x - x ^ { 2 }\). Given that \(x\) can vary,
3. find the value of \(x\) at which \(A\) is stationary,
4. find this stationary value of \(A\), and determine whether it is a maximum or a minimum value.

8
\includegraphics[max width=\textwidth, alt={}, center]{02da6b6a-6db1-4bc3-ad4e-537e4f61dcac-3_365_663_1813_740} The inside lane of a school running track consists of two straight sections each of length \(x\) metres, and two semicircular sections each of radius \(r\) metres, as shown in the diagram. The straight sections are perpendicular to the diameters of the semicircular sections. The perimeter of the inside lane is 400 metres.

1. Show that the area, \(A \mathrm {~m} ^ { 2 }\), of the region enclosed by the inside lane is given by \(A = 400 r - \pi r ^ { 2 }\).
2. Given that \(x\) and \(r\) can vary, show that, when \(A\) has a stationary value, there are no straight sections in the track. Determine whether the stationary value is a maximum or a minimum. [5]

4 Machines in a factory make cardboard cones of base radius \(r \mathrm {~cm}\) and vertical height \(h \mathrm {~cm}\). The volume, \(V \mathrm {~cm} ^ { 3 }\), of such a cone is given by \(V = \frac { 1 } { 3 } \pi r ^ { 2 } h\). The machines produce cones for which \(h + r = 18\).

1. Show that \(V = 6 \pi r ^ { 2 } - \frac { 1 } { 3 } \pi r ^ { 3 }\).
2. Given that \(r\) can vary, find the non-zero value of \(r\) for which \(V\) has a stationary value and show that the stationary value is a maximum.
3. Find the maximum volume of a cone that can be made by these machines.

3
\includegraphics[max width=\textwidth, alt={}, center]{2d5f452d-f820-40fc-9e22-9d3ac4f0698b-04_540_554_260_792} The diagram shows part of the curve \(y = x \left( 9 - x ^ { 2 } \right)\) and the line \(y = 5 x\), intersecting at the origin \(O\) and the point \(R\). Point \(P\) lies on the line \(y = 5 x\) between \(O\) and \(R\) and the \(x\)-coordinate of \(P\) is \(t\). Point \(Q\) lies on the curve and \(P Q\) is parallel to the \(y\)-axis.

1. Express the length of \(P Q\) in terms of \(t\), simplifying your answer.
2. Given that \(t\) can vary, find the maximum value of the length of \(P Q\).