Optimise geometric shape surface area/volume

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.

5 \includegraphics[max width=\textwidth, alt={}, center]{1a0e0afb-81be-45d1-8c86-f98e508e9a49-06_371_339_255_251} For a cone with base radius \(r\), height \(h\) and slant height \(l\), the following formulae are given.
Curved surface area, \(S = \pi r l\) Volume, \(V = \frac { 1 } { 3 } \pi r ^ { 2 } h\) A container is to be designed in the shape of an inverted cone with no lid. The base radius is \(r \mathrm {~m}\) and the volume is \(V \mathrm {~m} ^ { 3 }\). The area of the material to be used for the cone is \(4 \pi \mathrm {~m} ^ { 2 }\).

1. Show that \(V = \frac { 1 } { 3 } \pi \sqrt { 16 r ^ { 2 } - r ^ { 6 } }\).
2. In this question you must show detailed reasoning. It is given that \(V\) has a maximum value for a certain value of \(r\).
   Find the maximum value of \(V\), giving your answer correct to 3 significant figures.

\includegraphics{figure_4} Figure 4 shows a solid brick in the shape of a cuboid measuring 2x cm by x cm by y cm. The total surface area of the brick is 600 cm².

1. Show that the volume, \(V\) cm³, of the brick is given by \(V = 200x - \frac{4x^3}{3}\). [4]

Given that \(x\) can vary,

1. use calculus to find the maximum value of \(V\), giving your answer to the nearest cm³. [5]
2. Justify that the value of \(V\) you have found is a maximum. [2]

\includegraphics{figure_4} Figure 4 shows an open-topped water tank, in the shape of a cuboid, which is made of sheet metal. The base of the tank is a rectangle \(x\) metres by \(y\) metres. The height of the tank is \(x\) metres. The capacity of the tank is 100 m³.

1. Show that the area \(A\) m² of the sheet metal used to make the tank is given by $$A = \frac{300}{x} + 2x^2.$$ [4]
2. Use calculus to find the value of \(x\) for which \(A\) is stationary. [4]
3. Prove that this value of \(x\) gives a minimum value of \(A\). [2]
4. Calculate the minimum area of sheet metal needed to make the tank. [2]

\includegraphics{figure_7} Fig. 10 shows a solid cuboid with square base of side \(x\) cm and height \(h\) cm. Its volume is \(120\) cm\(^3\).

1. Find \(h\) in terms of \(x\). Hence show that the surface area, \(A\) cm\(^2\), of the cuboid is given by $$A = 2x^2 + \frac{480}{x}.$$ [3]
2. Find \(\frac{dA}{dx}\) and \(\frac{d^2A}{dx^2}\). [4]
3. Hence find the value of \(x\) which gives the minimum surface area. Find also the value of the surface area in this case. [5]

\includegraphics{figure_2} Figure 2 shows a solid cuboid \(ABCDEFGH\). \(AB = x\) cm, \(BC = 2x\) cm, \(AE = h\) cm The total surface area of the cuboid is 180 cm\(^2\). The volume of the cuboid is \(V\) cm\(^3\).

1. Show that \(V = 60x - \frac{4x^3}{3}\) [4]

Given that \(x\) can vary,

1. use calculus to find, to 3 significant figures, the value of \(x\) for which \(V\) is a maximum. Justify that this value of \(x\) gives a maximum value of \(V\). [5]
2. Find the maximum value of \(V\), giving your answer to the nearest cm\(^3\). [2]