In Exercises 4548, use De Moivres Theorem to find the indicated power of the complex

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. In Exercises \(\mathbf{u}=\langle 2,-1\rangle, \mathbf{v}=\langle 4,2\rangle, \text { and } \mathbf{w}=\langle 1,-3\rangle\) be vectors. Find the indicated expression. \(\mathbf{u}-\mathbf{v}\)

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. In Exercises \(\mathbf{u}=\langle 2,-1\rangle, \mathbf{v}=\langle 4,2\rangle, \text { and } \mathbf{w}=\langle 1,-3\rangle\) be vectors. Find the indicated expression. \(2 \mathbf{u}-3 \mathbf{w}\)

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. In Exercises \(\mathbf{u}=\langle 2,-1\rangle, \mathbf{v}=\langle 4,2\rangle, \text { and } \mathbf{w}=\langle 1,-3\rangle\) be vectors. Find the indicated expression. \(|\mathbf{u}+\mathbf{v}|\)

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. In Exercises \(\mathbf{u}=\langle 2,-1\rangle, \mathbf{v}=\langle 4,2\rangle, \text { and } \mathbf{w}=\langle 1,-3\rangle\) be vectors. Find the indicated expression. \(|\mathbf{w}-2 \mathbf{u}|\)

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. In Exercises \(\mathbf{u}=\langle 2,-1\rangle, \mathbf{v}=\langle 4,2\rangle, \text { and } \mathbf{w}=\langle 1,-3\rangle\) be vectors. Find the indicated expression. \(\mathbf{u} \cdot \mathbf{v}\)

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. In Exercises \(\mathbf{u}=\langle 2,-1\rangle, \mathbf{v}=\langle 4,2\rangle, \text { and } \mathbf{w}=\langle 1,-3\rangle\) be vectors. Find the indicated expression. \(\mathbf{u} \cdot \mathbf{w}\)

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. In Exercises 7–10, let \(A=(2,-1), B=(3,1), C=(-4,2)\) and \(D=(1,-5)\). Find the component form and magnitude of the vector. \(3 \overrightarrow{A B}\)

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. In Exercises 7–10, let \(A=(2,-1), B=(3,1), C=(-4,2)\) and \(D=(1,-5)\). Find the component form and magnitude of the vector. \(\overrightarrow{A B}+\overrightarrow{C D}\)

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. In Exercises 7–10, let \(A=(2,-1), B=(3,1), C=(-4,2)\) and \(D=(1,-5)\). Find the component form and magnitude of the vector. \(\overrightarrow{A C}+\overrightarrow{B D}\)

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. In Exercises 7–10, let \(A=(2,-1), B=(3,1), C=(-4,2)\) and \(D=(1,-5)\). Find the component form and magnitude of the vector. \(\overrightarrow{C D}+\overrightarrow{A B}\)

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. In Exercises 11 and 12, find (a) a unit vector in the direction of \(\overrightarrow{A B}\) and (b) a vector of magnitude 3 in the opposite direction. \(A=(4,0), B=(2,1)\)

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. In Exercises 11 and 12, find (a) a unit vector in the direction of \(\overrightarrow{A B}\) and (b) a vector of magnitude 3 in the opposite direction. \(A=(3,1), B=(5,1)\)

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. In Exercises 13 and 14, find (a) the direction angles of u and v and (b) the angle between u and v. \(\mathbf{u}=\langle 4,3\rangle, \mathbf{v}=\langle 2,5\rangle\)

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. In Exercises 13 and 14, find (a) the direction angles of u and v and (b) the angle between u and v. \(\mathbf{u}=\langle-2,4\rangle, \mathbf{v}=\langle 6,4\rangle\)

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. Use an algebraic method in Exercises 15–18 to convert the polar coordinates to rectangular coordinates. Approximate exact values with a calculator when appropriate. \(\left(-2.5,25^{\circ}\right)\)

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. Use an algebraic method in Exercises 15–18 to convert the polar coordinates to rectangular coordinates. Approximate exact values with a calculator when appropriate. \(\left(-3.1,135^{\circ}\right)\)

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. Use an algebraic method in Exercises 15–18 to convert the polar coordinates to rectangular coordinates. Approximate exact values with a calculator when appropriate. \((2,-\pi / 4)\)

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. Use an algebraic method in Exercises 15–18 to convert the polar coordinates to rectangular coordinates. Approximate exact values with a calculator when appropriate. \((3.6,3 \pi / 4)\)

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. In Exercises 19 and 20, polar coordinates of point P are given. Find all of its polar coordinates. \(P=(-1,-2 \pi / 3)\)

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. In Exercises 19 and 20, polar coordinates of point P are given. Find all of its polar coordinates. \(P=(-2,5 \pi / 6)\)

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. Use an algebraic method in Exercises 21–24 to find the polar coordinates of the given rectangular coordinates of point P that satisfy the stated conditions. Approximate exact values with a calculator when appropriate. \(\begin{array}{l}\text{(a) } 0 \leq \theta \leq 2 \pi\\\text{(b) } -\pi \leq \theta \leq \pi\\\text{(c) } 0 \leq \theta \leq 4 \pi\end{array}\) \(P=(2,-3)\)

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. Use an algebraic method in Exercises 21–24 to find the polar coordinates of the given rectangular coordinates of point P that satisfy the stated conditions. Approximate exact values with a calculator when appropriate. \(\begin{array}{l}\text{(a) } 0 \leq \theta \leq 2 \pi\\\text{(b) } -\pi \leq \theta \leq \pi\\\text{(c) } 0 \leq \theta \leq 4 \pi\end{array}\) \(P=(-10,0)\)

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. Use an algebraic method in Exercises 21–24 to find the polar coordinates of the given rectangular coordinates of point P that satisfy the stated conditions. Approximate exact values with a calculator when appropriate. \(\begin{array}{l}\text{(a) } 0 \leq \theta \leq 2 \pi\\\text{(b) } -\pi \leq \theta \leq \pi\\\text{(c) } 0 \leq \theta \leq 4 \pi\end{array}\) \(P=(5,0)\)

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. Use an algebraic method in Exercises 21–24 to find the polar coordinates of the given rectangular coordinates of point P that satisfy the stated conditions. Approximate exact values with a calculator when appropriate. \(\begin{array}{l}\text{(a) } 0 \leq \theta \leq 2 \pi\\\text{(b) } -\pi \leq \theta \leq \pi\\\text{(c) } 0 \leq \theta \leq 4 \pi\end{array}\) \(P=(0,-2)\)

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. In Exercises 25–30, eliminate the parameter t and identify the graph. Support your answer with a grapher. \(x=3-5 t, y=4+3 t\)

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. In Exercises 25–30, eliminate the parameter t and identify the graph. Support your answer with a grapher. \(x=4+t, y=-8-5 t,-3 \leq t \leq 5\)

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. In Exercises 25–30, eliminate the parameter t and identify the graph. Support your answer with a grapher. \(x=2 t^{2}+3, y=t-1\)

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. In Exercises 25–30, eliminate the parameter t and identify the graph. Support your answer with a grapher. \(x=3 \cos t, y=3 \sin t\)

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. In Exercises 25–30, eliminate the parameter t and identify the graph. Support your answer with a grapher. \(x=e^{2 t}-1, y=e^{t}\)

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. In Exercises 25–30, eliminate the parameter t and identify the graph. Support your answer with a grapher. \(x=t^{3}, y=\ln t, t>0\)

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. In Exercises 31 and 32, find a parametrization for the curve. The line through the points \((-1,-2) \text { and }(3,4)\)

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. In Exercises 31 and 32, find a parametrization for the curve. The line segment with endpoints \((-2,3) \text { and }(5,1)\)

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. Exercises 33 and 34 refer to the complex number \(z_{1}\) shown in the figure. If \(z_{1}=a+b i\), find a, b, and \(\left|z_{1}\right|\).

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. Exercises 33 and 34 refer to the complex number \(z_{1}\) shown in the figure. Find the trigonometric form of \(z_{1}\).

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. In Exercises 35–38, write the complex number in standard form. \(6\left(\cos 30^{\circ}+i \sin 30^{\circ}\right)\)

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. In Exercises 35–38, write the complex number in standard form. \(3\left(\cos 150^{\circ}+i \sin 150^{\circ}\right)\)

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. In Exercises 35–38, write the complex number in standard form. \(2.5\left(\cos \frac{4 \pi}{3}+i \sin \frac{4 \pi}{3}\right)\)

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. In Exercises 35–38, write the complex number in standard form. \(4(\cos 2.5+i \sin 2.5)\)

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. Use an algebraic method in Exercises 39–42 to find the trigonometric form with \(0 \leq \theta<2 \pi\) for the given complex number. Then write three other possible trigonometric forms for the number. Approximate exact values with a calculator when appropriate. \(3-3 i\)

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. Use an algebraic method in Exercises 39–42 to find the trigonometric form with \(0 \leq \theta<2 \pi\) for the given complex number. Then write three other possible trigonometric forms for the number. Approximate exact values with a calculator when appropriate. \(-1+i \sqrt{2}\)

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. Use an algebraic method in Exercises 39–42 to find the trigonometric form with \(0 \leq \theta<2 \pi\) for the given complex number. Then write three other possible trigonometric forms for the number. Approximate exact values with a calculator when appropriate. \(3-5 i\)

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. Use an algebraic method in Exercises 39–42 to find the trigonometric form with \(0 \leq \theta<2 \pi\) for the given complex number. Then write three other possible trigonometric forms for the number. Approximate exact values with a calculator when appropriate. \(-2-2 i\)

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. In Exercises 43 and 44, write the complex numbers \(z_{1} \cdot z_{2} \text { and } z_{1} / z_{2}\) in trigonometric form.. \(z_{1}=3\left(\cos 30^{\circ}+i \sin 30^{\circ}\right) \text { and } z_{2}=4\left(\cos 60^{\circ}+i \sin 60^{\circ}\right)\)

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. In Exercises 43 and 44, write the complex numbers \(z_{1} \cdot z_{2} \text { and } z_{1} / z_{2}\) in trigonometric form.. \(z_{1}=5\left(\cos 20^{\circ}+i \sin 20^{\circ}\right) \text { and } z_{2}=-2\left(\cos 45^{\circ}+i \sin 45^{\circ}\right)\)

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. In Exercises 45–48, use De Moivre’s Theorem to find the indicated power of the complex number. Write your answer in (a) trigonometric form and (b) standard form. \(\left[3\left(\cos \frac{\pi}{4}+i \sin \frac{\pi}{4}\right)\right]^{5}\)

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. In Exercises 45–48, use De Moivre’s Theorem to find the indicated power of the complex number. Write your answer in (a) trigonometric form and (b) standard form. \(\left[2\left(\cos \frac{\pi}{12}+i \sin \frac{\pi}{12}\right)\right]^{8}\)

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. In Exercises 45–48, use De Moivre’s Theorem to find the indicated power of the complex number. Write your answer in (a) trigonometric form and (b) standard form. \(\left[5\left(\cos \frac{5 \pi}{3}+i \sin \frac{5 \pi}{3}\right)\right]^{3}\)

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. In Exercises 45–48, use De Moivre’s Theorem to find the indicated power of the complex number. Write your answer in (a) trigonometric form and (b) standard form. \(\left[7\left(\cos \frac{\pi}{24}+i \sin \frac{\pi}{24}\right)\right]^{6}\)

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. In Exercises 49–52, find and graph the nth roots of the complex number for the specified value of n. \(3+3 i, \quad n=4\)

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. In Exercises 49–52, find and graph the nth roots of the complex number for the specified value of n. \(8, \quad n=3\)

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. In Exercises 49–52, find and graph the nth roots of the complex number for the specified value of n. \(1, \quad n=5\)

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. In Exercises 49–52, find and graph the nth roots of the complex number for the specified value of n. \(-1, \quad n=6\)

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. In Exercises 53–60, decide whether the graph of the given polar equation appears among the four graphs shown. \(r=3 \sin 4 \theta\)

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. In Exercises 53–60, decide whether the graph of the given polar equation appears among the four graphs shown. \(r=2+\sin \theta\)

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. In Exercises 53–60, decide whether the graph of the given polar equation appears among the four graphs shown. \(r=2+2 \sin \theta\)

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. In Exercises 53–60, decide whether the graph of the given polar equation appears among the four graphs shown. \(r=3|\sin 3 \theta|\)

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. In Exercises 53–60, decide whether the graph of the given polar equation appears among the four graphs shown. \(r=2-2 \sin \theta\)

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. In Exercises 53–60, decide whether the graph of the given polar equation appears among the four graphs shown. \(r=1-2 \cos \theta\)

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. In Exercises 53–60, decide whether the graph of the given polar equation appears among the four graphs shown. \(r=3 \cos 5 \theta\)

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. In Exercises 53–60, decide whether the graph of the given polar equation appears among the four graphs shown. \(r=3-2 \tan \theta\)

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. In Exercises 61–64, convert the polar equation to rectangular form and identify the graph. \(r=-2\)

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. In Exercises 61–64, convert the polar equation to rectangular form and identify the graph. \(r=-2 \sin \theta\)

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. In Exercises 61–64, convert the polar equation to rectangular form and identify the graph. \(r=-3 \cos \theta-2 \sin \theta\)

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. In Exercises 61–64, convert the polar equation to rectangular form and identify the graph. \(r=3 \sec \theta\)

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. In Exercises 65–68, convert the rectangular equation to polar form. Graph the polar equation. \(y=-4\)

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. In Exercises 65–68, convert the rectangular equation to polar form. Graph the polar equation. \(x=5\)

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. In Exercises 65–68, convert the rectangular equation to polar form. Graph the polar equation. \((x-3)^{2}+(y+1)^{2}=10\)

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. In Exercises 65–68, convert the rectangular equation to polar form. Graph the polar equation. \(2 x-3 y=4\)

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. In Exercises 69–72, analyze the graph of the polar curve. \(r=2-5 \sin \theta\)

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. In Exercises 69–72, analyze the graph of the polar curve. \(r=4-4 \cos \theta\)

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. In Exercises 69–72, analyze the graph of the polar curve. \(r=2 \sin 3 \theta\)

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. In Exercises 69–72, analyze the graph of the polar curve. \(r^{2}=2 \sin 2 \theta, 0 \leq \theta \leq 2 \pi\)

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. Graphing Lines Using Polar Equations (a) Explain why \(r=a \sec \theta\) is a polar form for the line \(x=a\). (b) Explain why \(r=b \csc \theta\) is a polar form for the line \(y=b\). (c) Let \(y=m x+b\). Prove that \(r=\frac{b}{\sin \theta-m \cos \theta}\) is a polar form for the line. What is the domain of r? (d) Illustrate the result in part (c) by graphing the line \(y=2 x+3\) using the polar form from part (c).

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. Flight Engineering An airplane is flying on a bearing of 80° at 540 mph. A wind is blowing with the bearing 100° at 55 mph. (a) Find the component form of the velocity of the airplane. (b) Find the actual speed and direction of the airplane.

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. Flight Engineering An airplane is flying on a bearing of 285° at 480 mph. A wind is blowing with the bearing 265° at 30 mph. (a) Find the component form of the velocity of the airplane. (b) Find the actual speed and direction of the airplane.

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. Combining Forces A force of 120 lb acts on an object at an angle of 20°. A second force of 300 lb acts on the object at an angle of \(-5^{\circ}\). Find the direction and magnitude of the resultant force.

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. Braking Force A 3000-pound car is parked on a street that makes an angle of 16° with the horizontal (see figure). (a) Find the force required to keep the car from rolling down the hill. (b) Find the component of the force perpendicular to the street.

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. Work Find the work done by a force F of 36 pounds acting in the direction given by the vector \(\langle 3,5\rangle\) in moving an object 10 feet from (0, 0) to (10, 0).

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. Height of an Arrow Stewart shoots an arrow straight up from the top of a building with initial velocity of 245 ft/sec. The arrow leaves from a point 200 ft above level ground. (a) Write an equation that models the height of the arrow as a function of time t. (b) Use parametric equations to simulate the height of the arrow. (c) Use parametric equations to graph height against time. (d) How high is the arrow after 4 sec? (e) What is the maximum height of the arrow? When does it reach its maximum height? (f) How long will it be before the arrow hits the ground?

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. Ferris Wheel Problem Lucinda is on a Ferris wheel of radius 35 ft that turns at the rate of one revolution every 20 sec. The lowest point of the Ferris wheel (6 o’clock) is 15 ft above ground level at the point (0, 15) of a rectangular coordinate system. Find parametric equations for the position of Lucinda as a function of time t in seconds if Lucinda starts \((t=0)\) at the point (35, 50).

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. Ferris Wheel Problem The lowest point of a Ferris wheel (6 o’clock) of radius 40 ft is 10 ft above the ground, and the center is on the y-axis. Find parametric equations for Henry’s position as a function of time t in seconds if his starting position \((t=0)\) is the point (0, 10) and the wheel turns at the rate of one revolution every 15 sec.

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. Ferris Wheel Problem Sarah rides the Ferris wheel described in Exercise 81. Find parametric equations for Sarah’s position as a function of time t in seconds if her starting position \((t=0)\) is the point (0, 90) and the wheel turns at the rate of one revolution every 18 sec.

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. Epicycloid The graph of the parametric equations \(x=4 \cos t-\cos 4 t, \quad y=4 \sin t-\sin 4 t\) is an epicycloid. The graph is the path of a point P on a circle of radius 1 rolling along the outside of a circle of radius 3, as suggested in the figure. (a) Graph simultaneously this epicycloid and the circle of radius 3. (b) Suppose the large circle has a radius of 4. Experiment! How do you think the equations in part (a) should be changed to obtain defining equations? What do you think the epicycloid would look like in this case? Check your guesses.

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. Throwing a Baseball Sharon releases a baseball 4 ft above the ground with an initial velocity of 66 ft/sec at an angle of 5° with the horizontal. How many seconds after the ball is thrown will it hit the ground? How far from Sharon will the ball be when it hits the ground?

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. Throwing a Baseball Diego releases a baseball 3.5 ft above the ground with an initial velocity of 66 ft/sec at an angle of 12° with the horizontal. How many seconds after the ball is thrown will it hit the ground? How far from Diego will the ball be when it hits the ground?

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. Field Goal Kicking Spencer practices kicking field goals 40 yd from a goal post with a crossbar 10 ft high. If he kicks the ball with an initial velocity of 70 ft/sec at a 45° angle with the horizontal (see figure), will Spencer make the field goal if the kick sails “true”?

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. Hang Time An NFL place-kicker kicks a football downfield with an initial velocity of 85 ft/sec. The ball leaves his foot at the 15 yard line at an angle of 56° with the horizontal. Determine the following: (a) The ball’s maximum height above the field. (b) The “hang time” (the total time the football is in the air).

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. Baseball Hitting Brian hits a baseball straight toward a 15-ft-high fence that is 400 ft from home plate. The ball is hit when it is 2.5 ft above the ground and leaves the bat at an angle of 30° with the horizontal. Find the initial velocity needed for the ball to clear the fence.

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. Throwing a Ball at a Ferris Wheel A 60-ft-radius Ferris wheel turns counterclockwise one revolution every 12 sec. Sam stands at a point 80 ft to the left of the bottom (6 o’clock) of the wheel. At the instant Kathy is at 3 o’clock, Sam throws a ball with an initial velocity of 100 ft/sec and an angle with the horizontal of 70°. He releases the ball from the same height as the bottom of the Ferris wheel. Find the minimum distance between the ball and Kathy.

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. Yard Darts Gretta and Lois are launching yard darts 20 ft from the front edge of a circular target of radius 18 in. If Gretta releases the dart 5 ft above the ground with an initial velocity of 20 ft/sec and at a 50° angle with the horizontal, will the dart hit the target?

Create a scatter plot for the data you collected or the data above.

With your calculator/computer in function mode, find values for a, b, c, and d so that the equation \(y=a \sin (b(x-c))+d\) (where y is distance and x is time) fits the distance versus time data plot.

Make a scatter plot of velocity versus time. Using the same a, b, and c values you found in (2), verify that the equation \(y=a b \cos (b(x-c))\) (where y is velocity and x is time) fits the velocity versus time data plot.

What do you think a plot of velocity versus distance (with velocity on the vertical axis and distance on the horizontal axis) would look like? Make a rough sketch of your prediction, then create a scatter plot of velocity versus distance. How well did your predicted graph match the actual data plot?

With your calculator/computer in parametric mode, graph the parametric curve \(x=a \sin (b(t-c))+d, y=a b \cos (b(t-c)), 0 \leq t \leq 2,\), where x represents distance, y represents velocity, and t is the time parameter. How well does this curve match the scatter plot of velocity versus time?