Chapter 10

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

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Write as a single logarithm: \(3 \log _{a} x+2 \log _{a} y-5 \log _{a} z\)

Graph each polar equation. $$ r=\cos \frac{\theta}{2} $$

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?

The letters \(x\) and \(y\) represent rectangular coordinates. Write each equation using polar coordinates \((r, \theta) .\) $$ r=2 $$

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Solve: \(\log _{5} \sqrt{x+4}=2\)

A force of magnitude 1200 pounds is required to prevent a car from rolling down a hill whose incline is \(15^{\circ}\) to the horizontal. What is the weight of the car?

The letters \(x\) and \(y\) represent rectangular coordinates. Write each equation using polar coordinates \((r, \theta) .\) $$ r=4 $$

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 \(a<0\)

A river has a constant current of \(3 \mathrm{~km} / \mathrm{h}\). At what angle to a boat dock should a motorboat capable of maintaining a constant speed of \(20 \mathrm{~km} / \mathrm{h}\) be headed in order to reach a point directly opposite the dock? If the river is \(\frac{1}{2}\) kilometer wide, how long will it take to cross?

The letters \(x\) and \(y\) represent rectangular coordinates. Write each equation using polar coordinates \((r, \theta) .\) $$ r=\frac{4}{1-\cos \theta} $$

Show that the graph of the equation \(r=2 a \sin \theta, a>0\) is a circle of radius \(a\) with center \((0, a)\) in rectangular coordinates.

The letters \(x\) and \(y\) represent rectangular coordinates. Write each equation using polar coordinates \((r, \theta) .\) $$ r=\frac{3}{3-\cos \theta} $$

Show that the graph of the equation \(r=-2 a \sin \theta, a>0\) is a circle of radius \(a\) with center \((0,-a)\) in rectangular coordinates.

A helicopter pilot needs to travel to a regional airport 25 miles away. She flies at an actual heading of \(\mathrm{N} 16.26^{\circ} \mathrm{E}\) with an airspeed of \(120 \mathrm{mph},\) and there is a wind blowing directly east at \(20 \mathrm{mph}\). (a) Determine the compass heading that the pilot needs to reach her destination. (b) How long will it take her to reach her destination? Round to the nearest minute.

In Chicago, the road system is set up like a Cartesian plane, where streets are indicated by the number of blocks they are from Madison Street and State Street. For example, Wrigley Field in Chicago is located at 1060 West Addison, which is 10 blocks west of State Street and 36 blocks north of Madison Street. Treat the intersection of Madison Street and State Street as the origin of a coordinate system, with east being the positive \(x\) -axis. (a) Write the location of Wrigley Field using rectangular coordinates. (b) Write the location of Wrigley Field using polar coordinates. Use the east direction for the polar axis. Express \(\theta\) in degrees. (c) Guaranteed Rate Field, home of the White \(\operatorname{Sox},\) is located at 35 th and Princeton, which is 3 blocks west of State Street and 35 blocks south of Madison. Write the location of Guaranteed Rate Field using rectangular coordinates. (d) Write the location of Guaranteed Rate Field using polar coordinates. Use the east direction for the polar axis. Express \(\theta\) in degrees.

Show that the graph of the equation \(r=2 a \cos \theta, a>0,\) is a circle of radius \(a\) with center \((a, 0)\) in rectangular coordinates.

Show that the graph of the equation \(r=-2 a \cos \theta, a>0\) is a circle of radius \(a\) with center \((-a, 0)\) in rectangular coordinates.

At 10: 15 A.M., a radar station detects an aircraft at a point 80 miles away and 25 degrees north of due east. At 10: 25 A.M., the aircraft is 110 miles away and 5 degrees south of due east. (a) Using the radar station as the pole and due east as the polar axis, write the two locations of the aircraft in polar coordinates. (b) Write the two locations of the aircraft in rectangular coordinates. Round answers to two decimal places. (c) What is the speed of the aircraft in miles per hour? Round the answer to one decimal place.

Radar station \(A\) uses a coordinate system where \(A\) is located at the pole and due east is the polar axis. On this system, two other radar stations, \(B\) and \(C,\) are located at coordinates \(\left(150,-24^{\circ}\right)\) and \(\left(100,32^{\circ}\right)\) respectively. If radar station \(B\) uses a coordinate system where \(B\) is located at the pole and due east is the polar axis, then what are the coordinates of radar stations \(A\) and \(C\) on this second system? Round answers to one decimal place.

In converting from polar coordinates to rectangular coordinates, what equations will you use?

Express \(r^{2}=\cos (2 \theta)\) in rectangular coordinates free of radicals.

Explain how to convert from rectangular coordinates to polar coordinates.

Prove that the area of the triangle with vertices \((0,0),\left(r_{1}, \theta_{1}\right),\) and \(\left(r_{2}, \theta_{2}\right), 0 \leq \theta_{1}<\theta_{2} \leq \pi,\) is $$ K=\frac{1}{2} r_{1} r_{2} \sin \left(\theta_{2}-\theta_{1}\right) $$

A 2-pound weight is attached to a 3 -pound weight by a rope that passes over an ideal pulley. The smaller weight hangs vertically, while the larger weight sits on a frictionless inclined ramp with angle \(\theta .\) The rope exerts a tension force \(\mathbf{T}\) on both weights along the direction of the rope. Find the angle measure for \(\theta\) that is needed to keep the larger weight from sliding down the ramp. Round your answer to the nearest tenth of a degree.

Solve: \(\log _{4}(x+3)-\log _{4}(x-1)=2\).

Use Descartes' Rule of Signs to determine the possible number of positive or negative real zeros for the function $$ f(x)=-2 x^{3}+6 x^{2}-7 x-8 $$

Find the midpoint of the line segment connecting the points (-3,7) and \(\left(\frac{1}{2}, 2\right)\).

Explain why the vertical-line test used to identify functions in rectangular coordinates does not work for equations expressed in polar coordinates.

Given that the point (3,8) is on the graph of \(y=f(x)\) what is the corresponding point on the graph of \(y=-2 f(x+3)+5 ?\)

If \(z=2-5 i\) and \(w=4+i,\) find \(z \cdot w\).

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Convert \(\frac{7 \pi}{3}\) radians to degrees.

Solve the equation: \(4 \sin \theta \cos \theta=1,0 \leq \theta<2 \pi\).

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Determine the amplitude and period of \(y=-2 \sin (5 x)\) without graphing.

Refer to Problem \(99 .\) The points \((-3,0),(-1,-2),(3,1),\) and (1,3) are the vertices of a parallelogram \(A B C D\). (a) Find the vertices of a new parallelogram \(A^{\prime} B^{\prime} C^{\prime} D^{\prime}\) if \(A B C D\) is translated by \(\mathbf{v}=\langle 3,-2\rangle\) (b) Find the vertices of a new parallelogram \(A^{\prime} B^{\prime} C^{\prime} D^{\prime}\) if \(A B C D\) is translated by \(-\frac{1}{2} \mathbf{v}\)

Solve the triangle: \(A=65^{\circ}, B=37^{\circ}, c=10\).

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Find any asymptotes for the graph of $$ R(x)=\frac{x+3}{x^{2}-x-12} $$

Find the exact value of \(\sin \frac{7 \pi}{12}\).

Simplify: \(\frac{5 x^{2} \cdot 3 e^{3 x}-e^{3 x} \cdot 10 x}{\left(5 x^{2}\right)^{2}}\)

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Find the area of a triangle with sides \(6,11,\) and \(13 .\)

Show that \(\sin ^{5} x=\sin x-2 \cos ^{2} x \sin x+\cos ^{4} x \sin x\).

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Solve: \(3^{2 x-3}=9^{1-x}\)

Explain in your own words what a vector is. Give an example of a vector.

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. \(m=f^{\prime}(x)=3 x^{2}+8 x\) gives the slope of the tangent line to the graph of \(f(x)=x^{3}+4 x^{2}-5\) at any number \(x\). Find an equation of the tangent line to \(f\) at \(x=-2\).

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Show that \(\cos ^{3} x=\cos x-\sin ^{2} x \cos x\).

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. $$ \text { Solve: } \sqrt[3]{x-2}=3 $$

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Factor \(-3 x^{3}+12 x^{2}+36 x\) completely.

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Find the exact value of \(\tan \left[\cos ^{-1}\left(\frac{1}{2}\right)\right]\)

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Find the amplitude, period, and phase shift of $$ y=\frac{3}{2} \cos (6 x+3 \pi) $$

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Find the distance between the points (-5,-8) and (7,1) .