SmarterEd

Aussie Maths & Science Teachers: Save your time with SmarterEd

Vectors, EXT2 V1 2023 HSC 15c

A curve \( \mathcal{C}\) spirals 3 times around the sphere centred at the origin and with radius 3, as shown.

A particle is initially at the point \((0,0,-3)\) and moves along the curve \(\mathcal{C}\) on the surface of the sphere, ending at the point \((0,0,3)\).
 

By using the diagram below, which shows the graphs of the functions  \(f(x)=\cos (\pi x)\)  and  \(g(x)=\sqrt{9-x^2}\), and considering the graph  \(y=f(x)g(x)\), give a possible set of parametric equations that describe the curve \( \mathcal{C}\).  (3 marks)
 

Show Answers Only

\(x= \cos{(\pi t)}\sqrt{9-t^2} \)

\(y= -\sin{(\pi t)}\sqrt{9-t^2} \)

\( z=t \)

Show Worked Solution

\(\text{Since the curve lies on a sphere with radius 3:}\)

\(x^2+y^2+z^2=3^3 \)

\(\text{Considering the graph}\ \ y=\cos (\pi t)\sqrt{9-t^2}\ \ \text{(as per hint)} \)

\(\Big(\cos (\pi t)\sqrt{9-t^2}\Big)^2+\Big(\sin (\pi t)\sqrt{9-t^2}\Big)^2+t^2=3^2 \ \ …\ (1) \)

\(\text{Since}\ z\ \text{increases and}\ x\ \text{and}\ y\ \text{change signs} \)

\( \Rightarrow z=t \)
 

\(\text{In order to satisfy the equation in (1): } \)

\( x,y\ \text{must be one of }\ \ \pm \cos{(\pi t)}\sqrt{9-t^2}\ \ \text{or}\ \ \pm \sin{(\pi t)}\sqrt{9-t^2} \)
 

\(\text{At}\ \ z=0,\ t=0, \ x=3\ \ \text{(from graph):} \)

\( \Rightarrow x= \cos{(\pi t)}\sqrt{9-t^2} \)
 

\(\text{At}\ \ z=0+\epsilon,\ t=0+\epsilon, \ y \lt 0\ \ \text{(from graph):} \)

\( \Rightarrow y= -\sin{(\pi t)}\sqrt{9-t^2} \)

♦♦♦ Mean mark 22%.