Problem 69
Listeners \(A(-8,0), B(8,0)\), and \(C(8,10)\) recorded the exact times at which they heard an explosion. If \(B\) and \(C\) heard the explosion at the same time and \(A\) heard it 12 seconds later, where was the explosion? Assume that distances are in kilometers and that sound travels \(\frac{1}{3}\) kilometer per second.
Problem 69
. Plot Lissajous figures for the following combinations of \(a\) and \(b\) for \(0 \leq t \leq 2 \pi\) : (a) \(a=1, b=2\) (b) \(a=4, b=8\) (c) \(a=5, b=10\) (d) \(a=2, b=3\) (e) \(a=6, b=9\) (f) \(a=12, b=18\)
Problem 70
Show that \(\left(\sqrt{x^{2}-a^{2}}-x\right) \rightarrow 0\) as \(x \rightarrow \infty\). Hint: Rationalize the numerator.
Problem 71
For an ellipse, let \(p\) and \(q\) be the distances from a focus to the two vertices. Show that \(b=\sqrt{p q}\), with \(2 b\) being the minor diameter.
Problem 71
Plot the following parametric curves. Describe in words how the point moves around the curve in each case. (a) \(x=\cos \left(t^{2}-t\right), y=\sin \left(t^{2}-t\right)\) (b) \(x=\cos \left(2 t^{2}+3 t+1\right), y=\sin \left(2 t^{2}+3 t+1\right)\) (c) \(x=\cos (-2 \ln t), y=\sin (-2 \ln t)\) (d) \(x=\cos (\sin t), y=\sin (\sin t)\)
Problem 72
. Using a computer algebra system, plot the following parametric curves for \(0 \leq t \leq 2 .\) Describe the shape of the curve in each case and the similarities and differences among all the curves. (a) \(x=t, y=t^{2}\) (b) \(x=t^{3}, y=t^{6}\) (c) \(x=-t^{4}, y=-t^{8}\) (d) \(x=t^{5}, y=t^{10}\)
Problem 73
Let \(P\) be a point on a ladder of length \(a+b, P\) being \(a\) units from the top end. As the ladder slides with its top end on the \(y\) -axis and its bottom end on the \(x\) -axis, \(P\) traces out a curve. Find the equation of this curve.
Problem 74
. Draw the graph of the epicycloid (see Problem 63 ) $$ \begin{array}{l} x=(a+b) \cos t-b \cos \frac{a+b}{b} t \\ y=(a+b) \sin t-b \sin \frac{a+b}{b} t \end{array} $$ for various values of \(a\) and \(b\). What conjectures can you make \((\) see Problem 73\() ?\)
Problem 75
Draw the Folium of Descartes \(x=3 t /\left(t^{3}+1\right)\), \(y=3 t^{2} /\left(t^{3}+1\right) .\) Then determine the values of \(t\) for which this graph is in each of the four quadrants.
Problem 75
If a horizontal hyperbola and a vertical hyperbola have the same asymptotes, show that their eccentricities \(e\) and \(E\) satisfy \(e^{-2}+E^{-2}=1\).
