Chapter 7

Find the components of the vector in standard position that satisfy the given conditions. Magnitude \(4.6 ;\) points due west

Find all the complex solutions of the equations. $$z^{3}+1=0$$

Leisure Malik, Keisha, and Brian get together for a game of pitch and catch. At a certain moment, Brian is 11 feet away from Malik and 9 feet away from Keisha, and the lines from Keisha to Malik and from Keisha to Brian form an angle of \(62^{\circ} .\) How far apart are Malik and Keisha?

In Exercises \(31-46,\) sketch the graphs of the polar equations. $$r=2+3 \cos \theta$$

Convert each of the given rectangular equations to polar form. $$x+2 y=4$$

Find the components of the vector in standard position that satisfy the given conditions. Length \(3.1 ;\) direction \(16^{\circ}\) south of east

Find all the complex solutions of the equations. $$z^{2}-i=0$$

Games A billiard ball traverses a distance of 26 inches on a straight-line path, and then it collides with another ball, changes direction, and traverses a distance of 18 inches on a different straight-line path before coming to a stop. If an angle of \(37^{\circ}\) is formed from the lines that connect the initial location of the ball to the final location of the ball and to the point of the collision, what are the two possible values of the distance \(d\) between the initial and final locations of the ball? Sketch a figure first.

In this set of exercises, you will use vectors and dot products to study real- world problems. Computer Animation An animated figure's location is given by \(\langle 5,2\rangle .\) By what angle must the figure be rotated so that its new location is in the direction of \langle 4,3\rangle ? Round your answer to the nearest tenth of a degree.

In Exercises \(31-46,\) sketch the graphs of the polar equations. $$r=2-3 \sin \theta$$

Convert each of the given rectangular equations to polar form. $$3 x+y=1$$

Find all the complex solutions of the equations. $$i z^{3}=1$$

In Exercises \(47-54,\) use a graphing utility to graph the polar equations. $$r=5-4 \sin \theta$$

Convert each of the given rectangular equations to polar form. $$x^{2}+y^{2}=25$$

The distance from the top of a utility pole to a certain point \(P\) on the surrounding level ground is 20 feet, and the angle of elevation of the top of the pole with respect to point \(P\) is \(38^{\circ} .\) What is the distance from the top of the pole to a point on the ground that is 10 feet farther away from the base of the pole than \(P ?\)

Find the components of the vector in standard position that satisfy the given conditions. Magnitude \(59 ;\) direction \(108^{\circ}\)

Find all the complex solutions of the equations. $$z^{3}+i z=0$$

Utilities A telephone pole is positioned beside a road that has a slope of \(10^{\circ}\) from the horizontal. When the angle of elevation of the sun is \(65^{\circ},\) the telephone pole casts a shadow that is 20.6 feet long. How tall is the telephone pole?

In this set of exercises, you will use vectors and dot products to study real- world problems. Work on Incline A box weighing 100 pounds is pushed up a hill. The hill makes an angle of \(30^{\circ}\) with the horizontal. Find the work done against gravity in pushing the box a distance of 60 feet. Round your answer to the nearest foot-pound.

In Exercises \(47-54,\) use a graphing utility to graph the polar equations. $$r=3+4 \cos \theta$$

Convert each of the given rectangular equations to polar form. $$x^{2}+y^{2}=4$$

Sarah and Joycelynn go for a hike. On the first leg, they walk 3.2 miles in the direction \(\mathrm{E} 13^{\circ} \mathrm{S}\). On the second and final leg, they walk 2.7 miles in the direction E56"S. At the end of the hike, how far are they from their starting point?

Round your answers to two decimal places. The world's largest weathervane is located in Montague, Michigan. On a July day in 2007 , it showed that the wind had a speed of 15 miles per hour in the direction \(S 30^{\circ} \mathrm{E}\). Express the wind velocity in component form. (Source: www. wunderground.com)

Let \(z=r(\cos \theta+i \sin \theta)\) be a nonzero complex number, and let \(n\) be a positive integer greater than 1. Verify that each of the following \(n\) numbers is a solution of the equation \(u^{n}=z:\) $$\begin{aligned} &\sqrt[n]{r}\left[\cos \left(\frac{\theta+2 \pi k}{n}\right)+i \sin \left(\frac{\theta+2 \pi k}{n}\right)\right]\\\ &k=0,1,2, \ldots, n-1 \end{aligned}$$ where \(\sqrt[n]{r}\) denotes the positive real number that, when raised to the \(n\) th power, gives \(r .\) (Hint: Use De Moivre's Theorem.)

In Exercises \(47-54,\) use a graphing utility to graph the polar equations. $$r=\cos \left(\theta-\frac{\pi}{4}\right)$$

Convert each of the given rectangular equations to polar form. $$(x+1)^{2}+y^{2}=1$$

A square is inscribed in a circle of radius 15 inches. Find the area of the square.

Round your answers to two decimal places. A golf ball is hit from a tee with a launch angle of \(13.2^{\circ}\) and speed 140 miles per hour. Express the velocity of the ball in component form. (Source: www.golf.com)

Can two or more of the \(n\) solutions of the equation \(u^{n}=z\) be equal?

In this set of exercises, you will use vectors and dot products to study real- world problems. Power The horsepower \(P\) of an engine pulling a cart is determined by the formula $$P=\frac{1}{550}(F \cdot v)$$ where \(F\) is the force, in pounds, exerted on the cart and \(v\) is the velocity, in feet per second, of the cart as it is moved by the engine. Find the horsepower of an engine that is exerting a force of 2000 pounds at an angle of \(30^{\circ}\) and is moving the cart horizontally at a speed of 15 feet per second. Round to the nearest tenth of a horsepower.

In Exercises \(47-54,\) use a graphing utility to graph the polar equations. $$r=\sin \left(\theta+\frac{\pi}{4}\right)$$

Convert each of the given rectangular equations to polar form. $$x^{2}+(y+3)^{2}=9$$

Round your answers to two decimal places. The Beaufort scale was developed in 1805 by Sir Francis Beaufort of England. It gives a measure for wind intensity based on observed sea and land conditions. For example, a wind speed of 20 knots is classified as a "fresh breeze," and smaller trees sway at this wind speed. Note that wind speed can also be measured in knots , where 1 knot equals 1.15 miles per hour. (Source: www.noaa.gov) (a) If the fresh breeze is in the direction \(\mathrm{S} 60^{\circ} \mathrm{W}\), express the velocity of the breeze in component form. Use knots for the unit of speed. (b) Express the velocity of the fresh breeze in component form using miles per hour as the unit for speed.

How many solutions of the equation \(u^{n}=z\) are real numbers if \(n\) is odd and \(z\) is real (that is, the imaginary part of \(z\) is zero)?

This set of exercises will draw on the ideas presented in this section and your general math background. Find \(a\) such that \(\langle 4, a\rangle\) and \langle-3,2\rangle are orthogonal.

In Exercises \(47-54,\) use a graphing utility to graph the polar equations. $$r=3 \sin 4 \theta$$

How many solutions of the equation \(u^{n}=z\) are real numbers if \(n\) is even and \(z\) is real (that is, the imaginary part of \(z\) is zero)?

Find \(a\) such that \(\langle 4, a\rangle\) and \langle-3,2\rangle are orthogonal. Prove that if \(\mathbf{v}\) and \(\mathbf{w}\) are nonzero orthogonal vectors, then proj.v \(=0\)

In Exercises \(47-54,\) use a graphing utility to graph the polar equations. $$r=2 \cos 3 \theta$$

Round your answers to two decimal places. Wanda goes for a hike. She first walks 2.4 miles in the direction \(S 17^{\circ} \mathrm{E}\) and then goes another 1.8 miles in the direction \(\mathrm{S} 38^{\circ} \mathrm{E}\). (a) By what east-west distance did Wanda's position change between the time she began the hike and the time she completed it? (b) By what north-south distance did Wanda's position change? (c) At the end of the hike, how far is Wanda from her starting point? (d) Suppose that Wanda traverses a single straight-line path and that her starting point and ending point are the same as before. In what direction does she walk?

This set of exercises will draw on the ideas presented in this section and your general math background. Prove the following for vectors \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w}: \quad \mathbf{u} \cdot(\mathbf{v}+\mathbf{w})=\) \(\mathbf{u} \cdot \mathbf{v}+\mathbf{u} \cdot \mathbf{w}\)

In Exercises \(47-54,\) use a graphing utility to graph the polar equations. $$r=2 \theta, 0 \leq \theta \leq 4 \pi$$

An archer shoots two arrows at a target. The second arrow lands twice as far from the center of the target as the first arrow. The points at which the arrows hit the target are 6.5 inches apart, and an angle of \(76^{\circ}\) is formed by the line segments that connect the center of the target to those two points. How far from the center of the target does each of the arrows land?

Round your answers to two decimal places. A ball is thrown upward with a velocity of 20 meters per second at an angle of \(42^{\circ}\) with respect to the horizontal. (a) At the time the ball is thrown, how fast is it moving in the horizontal direction? (b) At the time the ball is thrown, how fast is it moving in the vertical direction?

This set of exercises will draw on the ideas presented in this section and your general math background. Prove the following for any vector \(\mathbf{u :} \quad 0 \cdot \mathbf{u}=0\)

In Exercises \(47-54,\) use a graphing utility to graph the polar equations. $$r=\frac{1}{\theta}, 0<\theta \leq 4 \pi$$

This set of exercises will draw on the ideas preEented in this section and your general math background. Prove that the Law of Cosines holds for a triangle that has an obtuse angle.

Prove the following for any vector \(\mathbf{u :} \quad 0 \cdot \mathbf{u}=0\) Prove the following for any vector \(\mathbf{v}: \quad \mathbf{v} \cdot \mathbf{v}=\|\mathbf{v}\|^{2} .\) (In advanced mathematics, this relationship is very useful.)

In Exercises \(55-58,\) use a graphing utility to find the smallest value of \(\theta\) max, with \(\theta \min =0,\) such that the entire curve is graphed exactly once without retracing. $$r=-2 \cos \theta$$

Convert each of the given polar equations to rectangular form. $$r=3$$