91Ó°ÊÓ

Find all the complex roots. Write your answers in exponential form. The complex fourth roots of \(\sqrt{3}-i\)

The rectangular coordinates of a point are given. Find polar coordinates for each point. $$ (5,5 \sqrt{3}) $$

Find the direction angle of \(\mathbf{v}\). \(\mathbf{v}=6 \mathbf{i}-4 \mathbf{j}\)

A man pushes a wheelbarrow up an incline of \(20^{\circ}\) with a force of 100 pounds. Express the force vector \(\mathbf{F}\) in terms of \(\mathbf{i}\) and \(\mathbf{j}\).

A Boeing 787 Dreamliner maintains a constant airspeed of 550 miles per hour (mph) headed due north. The jet stream is \(100 \mathrm{mph}\) in the northeasterly direction. (a) Express the velocity \(\mathbf{v}_{\mathrm{a}}\) of the 787 relative to the air and the velocity \(\mathbf{v}_{\mathrm{w}}\) of the jet stream in terms of \(\mathbf{i}\) and \(\mathbf{j}\). (b) Find the velocity of the 787 relative to the ground. (c) Find the actual speed and direction of the 787 relative to the ground.

The letters \(x\) and \(y\) represent rectangular coordinates. Write each equation using polar coordinates \((r, \theta) .\) $$ r=\sin \theta+1 $$

Airplane An airplane has an airspeed of 500 kilometers per hour \((\mathrm{km} / \mathrm{h})\) bearing \(\mathrm{N} 45^{\circ} \mathrm{E}\). The wind velocity is \(60 \mathrm{~km} / \mathrm{h}\) in the direction \(\mathrm{N} 30^{\circ} \mathrm{W}\). Find the resultant vector representing the path of the plane relative to the ground. What is the groundspeed of the plane? What is its direction?

An airplane has an airspeed of \(600 \mathrm{~km} / \mathrm{h}\) bearing \(\mathrm{S} 30^{\circ} \mathrm{E}\). The wind velocity is \(40 \mathrm{~km} / \mathrm{h}\) in the direction \(\mathrm{S} 45^{\circ} \mathrm{E}\). Find the resultant vector representing the path of the plane relative to the ground. What is the groundspeed of the plane? What is its direction?

A force of magnitude 700 pounds is required to hold a boat and its trailer in place on a ramp whose incline is \(10^{\circ}\) to the horizontal. What is the combined weight of the boat and its trailer?

Show that the graph of the equation \(r \cos \theta=a\) is a vertical line \(a\) units to the right of the pole if \(a \geq 0\) and \(|a|\) units to the left of the pole if \(a<0\)