5 A designer intends to manufacture a product using a 3-D printer. The product will take the form of a surface $S$ which must meet a number of design specifications. The designer chooses to model $S$ with the equation $\mathrm { Z } = \mathrm { y } \cosh \mathrm { x }$ for $- \ln 20 \leqslant x \leqslant \ln 20 , - 2 \leqslant y \leqslant 2$.
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item In the Printed Answer Booklet, on the axes provided, sketch the section of $S$ given by $y = 1$.
\item One of the design specifications of the product is that this section should have a length no greater than 20 units.

Determine whether the product meets this requirement according to the model.
\end{enumerate}\item \begin{enumerate}[label=(\roman*)]
\item In the Printed Answer Booklet, on the axes provided, sketch the contour of $S$ given by $z = 1$.
\item When this contour is rotated through $2 \pi$ radians about the $x$-axis, the surface $T$ is generated. The surface area of $T$ is denoted by $A$.

Show that $A$ can be written in the form $\mathrm { k } \pi \int _ { 0 } ^ { \ln 20 } \frac { 1 } { \cosh ^ { 3 } \mathrm { x } } \sqrt { \cosh ^ { 4 } \mathrm { x } + \cosh ^ { 2 } \mathrm { x } - 1 } \mathrm { dx }$ for some\\
integer $k$ to be determined. integer $k$ to be determined.
\item A second design specification is that the surface area of $T$ must not be greater than 20 square units.

Use your calculator to decide whether the product meets this requirement according to the model.
\end{enumerate}\end{enumerate}

\hfill \mbox{\textit{OCR Further Additional Pure 2020 Q5 [13]}}