8.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ffa0b566-6448-491b-96d7-d3806bcfe063-4_483_453_1503_623}
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\caption{Figure 3}
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A manufacturer produces cartons for fruit juice. Each carton is in the shape of a closed cuboid with base dimensions \(2 x \mathrm {~cm}\) by \(x \mathrm {~cm}\) and height \(h \mathrm {~cm}\), as shown in Fig. 3.
Given that the capacity of a carton has to be \(1030 \mathrm {~cm} ^ { 3 }\),
- express \(h\) in terms of \(x\),
- show that the surface area, \(A \mathrm {~cm} ^ { 2 }\), of a carton is given by \(A = 4 x ^ { 2 } + \frac { 3090 } { x }\).
The manufacturer needs to minimise the surface area of a carton.
- Use calculus to find the value of \(x\) for which \(A\) is a minimum.
- Calculate the minimum value of \(A\).
- Prove that this value of \(A\) is a minimum.