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 \(- \ln 20 \leqslant x \leqslant \ln 20 , - 2 \leqslant y \leqslant 2\).

1. 1. In the Printed Answer Booklet, on the axes provided, sketch the section of \(S\) given by \(y = 1\).
   2. 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.
2. 1. In the Printed Answer Booklet, on the axes provided, sketch the contour of \(S\) given by \(z = 1\).
   2. 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.
   3. 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.