91Ó°ÊÓ

Ground Speed and Bearing of a Plane A plane is headed due south with an air- speed of 192 mph. A wind from a direction of 78.0 \(^{\circ}\) is blowing at 23.0 mph. Find the ground speed and resulting bearing of the plane.

Write each complex number in rectangular form. 5 cis \(300^{\circ}\)

In rectangular coordinates, the graph of $$a x+b y=c$$ is a horizontal line if \(a=0\) or a vertical line if \(b=0\). Work Exercises in order, to determine the general forms of polar equations for horizontal and vertical lines. Sketch the graph of the equation $$r=3 \csc \theta.$$ What is the corresponding rectangular equation?

"The shortest distance between two points is a straight line." Explain how this is related to the geometric property that states that the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side.

Solve each triangle \(A B C\) that exists. See Examples \(4-6\) $$A=42.5^{\circ}, a=15.6 \mathrm{ft}, b=8.14 \mathrm{ft}$$

Airports \(A\) and \(B\) are \(450 \mathrm{km}\) apart, on an east-west line. Tom flies in a northeast direction from \(A\) to airport \(C .\) From \(C\) he flies \(359 \mathrm{km}\) on a bearing of \(128^{\circ} 40^{\prime}\) to \(B .\) How far is \(C\) from \(A ?\)

The Vietnam Veterans Memorial in Washington, D.C., is V-shaped with equal sides of length \(246.75 \mathrm{ft}\). The angle between these sides measures \(125^{\circ} 12^{\prime} .\) Find the distance between the ends of the two sides. (Source: Pamphlet obtained at Vietnam Veterans Memorial.)

Show that the hyperbolic spiral \(r \theta=a\), where \(r^{2}=x^{2}+y^{2}\), is given parametrically by $$x=\frac{a \cos \theta}{\theta}, \quad y=\frac{a \sin \theta}{\theta}, \quad \text { for } \theta \text { in }(-\infty, 0) \cup(0, \infty)$$

Radio direction finders are placed at points \(A\) and \(B\), which are 3.46 mi apart on an east-west line, with \(A\) west of \(B\). From \(A\) the bearing of a certain radio transmitter is \(47.7^{\circ},\) and from \(B\) the bearing is \(302.5^{\circ} .\) Find the distance of the transmitter from \(A\)

Explain why a real number can have only one real cube root.