Sketching surface sections

8 A surface, \(C\), is given by the equation \(z = \mathrm { f } ( x , y )\) for all real values of \(x\) and \(y\). You are given that \(C\) has the following properties.

• The surface is continuous for all \(x\) and \(y\).
• The contour \(z = - 1\) is a single point on the \(z\)-axis.
• For \(- 1 < a < 1\), the contour \(z = a\) is a pair of circles with different radiuses but each having the same centre \(( 0,0 , a )\).
• The contour \(z = 1\) consists of the circle, centre \(( 0,0,1 )\) and radius 1 .

Sketch a possible section of \(C\) corresponding to \(y = 0\).

1 The surface \(S\) is defined for all real \(x\) and \(y\) by the equation \(z = x ^ { 2 } + 2 x y\). The intersection of \(S\) with the plane \(\Pi\) gives a section of the surface. On the axes provided in the Printed Answer Booklet, sketch this section when the equation of \(\Pi\) is each of the following.

1. \(x = 1\)
2. \(y = 1\)

10 \includegraphics[max width=\textwidth, alt={}, center]{df94bc38-5187-4349-9005-f9b72691c70d-4_519_770_251_242} A student wishes to model the saddle of a horse. They use a surface described by a function of the form \(\mathrm { z } = \mathrm { f } ( \mathrm { x } , \mathrm { y } )\) with a saddle point at the origin \(O\). The z -axis is vertically upwards. The \(x\) - and \(y\)-axes lie in a horizontal plane, with the \(x\)-axis across the horse and the \(y\)-axis along the length of the horse (see diagram). The arc \(A O B\) is part of a parabola which lies in the \(y z\)-plane. The arc \(C O D\) is part of a parabola which lies in the \(x z\)-plane. The saddle is symmetric in both the \(x z\)-plane and \(y z\)-plane. The length of the saddle, the distance \(A B\), is to be 0.6 m with both \(A\) and \(B\) at a height of 0.27 m above \(O\). The width of the saddle, the distance \(C D\), is to be 0.5 m with both \(C\) and \(D\) at a depth of 0.4 m below \(O\).

1. On separate diagrams, sketch the sections \(x = 0\) and \(y = 0\).
   [0pt]
2. Determine a function f that describes the saddle. [You do not need to state the domain of function f .] \section*{END OF QUESTION PAPER} \section*{OCR
   Oxford Cambridge and RSA}