Chapter 7

Construction A homeowner constructs a wooden border for a triangular flower bed. One edge of the bed lies along a north-south line and is 12 feet long. A second edge begins at the southern tip of the first edge and is oriented \(65^{\circ}\) west of south. If the length of the third edge is 18 feet, what is the perimeter of the flower bed?

Perform each operation, given \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w}\) $$\mathbf{u}=\langle 3,2\rangle, \mathbf{v}=\langle-1,4\rangle, \mathbf{w}=\langle-2,-1\rangle$$ $$-2 \mathbf{w} \cdot(\mathbf{v}+\mathbf{w})$$

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

For each of the points given in polar coordinates, find two additional pairs of polar coordinates \((r, \theta),\) one with \(r>0\) and one with \(r<0\). $$(5,0)$$

A store has had a triangular sign made with its name on it. The edges of the sign are 11 inches, 14 inches, and 8 inches in length. Find the measure of the angle opposite each edge.

Find a unit vector in the same direction as the given vector. $$\mathbf{w}=\langle 1,1\rangle$$

Find the square roots of each complex number. Round all numbers to three decimal places. $$1+\sqrt{3} i$$

Hiking A group of people go hiking. On the first leg, they hike 2.5 miles due north. The direction of the second and final leg is \(\mathrm{N} 36^{\circ} \mathrm{E}\). If they end up at a place that is 5.8 miles from their starting point, how great a distance did they traverse? Sketch a figure first.

Perform each operation, given \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w}\) $$\mathbf{u}=\langle 3,2\rangle, \mathbf{v}=\langle-1,4\rangle, \mathbf{w}=\langle-2,-1\rangle$$ $$3 u+v-2 w$$

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

For each of the points given in polar coordinates, find two additional pairs of polar coordinates \((r, \theta),\) one with \(r>0\) and one with \(r<0\). $$\left(\frac{3}{4}, \frac{\pi}{6}\right)$$

Find a unit vector in the same direction as the given vector. $$\mathbf{u}=\langle 3,2\rangle$$

Find the square roots of each complex number. Round all numbers to three decimal places. $$-2-2 i$$

Distance A squirrel runs from Point A to Point B along am above-ground power line and notices a peanut on the ground. The angles of depression of the peanut with respect to Point \(A\) and Point \(B\) are \(39^{\circ}\) and \(54^{\circ}\), respectively. If those points are 9.1 feet apart and the peanut directly below the power line, how far above the ground is the squirrel?

Perform each operation, given \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w}\) $$\mathbf{u}=\langle 3,2\rangle, \mathbf{v}=\langle-1,4\rangle, \mathbf{w}=\langle-2,-1\rangle$$ $$-\mathbf{u}-2 \mathbf{v}+\mathbf{w}$$

In Exercises \(31-46,\) sketch the graphs of the polar equations. $$r=4 \cos (3 \theta)$$

Find a unit vector in the same direction as the given vector. $$\mathbf{v}=-2 \mathbf{i}+1 \mathbf{j}$$

Find the square roots of each complex number. Round all numbers to three decimal places. Find the fifth roots of -1

Archery An archer shoots two arrows at a target. The angle formed by the lines that connect the center of the target to the points at which the arrows hit the target is \(109^{\circ} .\) If those points are 15 inches apart and one of them is 7.3 inches from the center, how far from the center is the other point?

In Exercises \(31-46,\) sketch the graphs of the polar equations. $$r^{2}=4 \cos (2 \theta)$$

For each of the points given in polar coordinates, find two additional pairs of polar coordinates \((r, \theta),\) one with \(r>0\) and one with \(r<0\). $$\left(-\frac{11}{7},-5 \pi\right)$$

A walkway around a flower bed in a park is made up of three straight sections that form the sides of a triangle. If the lengths of the sides are 26 feet, 24 feet, and 21 feet, what is the angle opposite the longest side?

Find a unit vector in the same direction as the given vector. $$\mathbf{u}=4 \mathbf{i}-3 \mathbf{j}$$

Find the fifth roots of \(i\)

Perform each operation, given \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w}\) $$\mathbf{u}=\langle 3,2\rangle, \mathbf{v}=\langle-1,4\rangle, \mathbf{w}=\langle-2,-1\rangle$$ $$\operatorname{proj}_{u}(\mathbf{v}+\mathbf{w})$$

In Exercises \(31-46,\) sketch the graphs of the polar equations. $$r^{2}=9 \sin (2 \theta)$$

For each of the points given in polar coordinates, find two additional pairs of polar coordinates \((r, \theta),\) one with \(r>0\) and one with \(r<0\). $$\left(\frac{7}{2}, 3 \pi\right)$$

The lengths of the two sections of a hospital bed are 3 feet and 4 feet. What is the angle between the two sections of the bed when one section is raised up so that the tip of the head of the bed is 6 feet from the tip of the foot?

Find the components of the vector in standard position that satisfy the given conditions. Magnitude \(19 ;\) direction \(34^{\circ}\)

Find the fourth roots of -16

Geometry Marisa has a triangular sign made with her last name on it. She has the sign attached to her lamppost so that visitors can easily identify her house. The lengths of two edges of the sign are 10 inches and 7 inches, and the angle opposite the 10 -inch edge is \(75^{\circ} .\) What is the length of the third edge?

In this set of exercises, you will use vectors and dot products to study real- world problems. \- Work A parent pulling a wagon in which her child is riding along level ground exerts a force of 20 pounds on the handle. The handle makes an angle of \(45^{\circ}\) with the horizontal. How much work is done in pulling the wagon 100 feet, to the nearest foot-pound?

In Exercises \(31-46,\) sketch the graphs of the polar equations. $$r=4-3 \cos \theta$$

For each of the points given in polar coordinates, find two additional pairs of polar coordinates \((r, \theta),\) one with \(r>0\) and one with \(r<0\). $$\left(1.3, \frac{3 \pi}{4}\right)$$

The lengths of two edges of a triangular bandage are 8 inches and 5 inches, and the angle formed by those two edges is \(85^{\circ} .\) How long is the third edge of the bandage, and what is the area of the bandage?

Find the components of the vector in standard position that satisfy the given conditions. Length \(7 ;\) direction \(276^{\circ}\)

Find the cube roots of \(8 i\)

In this set of exercises, you will use vectors and dot products to study real- world problems. A child pulls a wagon along level ground. He exerts a force of 20 pounds on the handle, which makes a \(30^{\circ}\) angle with the horizontal. Find the work done in pulling the wagon 100 feet, to the nearest foot-pound.

In Exercises \(31-46,\) sketch the graphs of the polar equations. $$r=5+4 \sin \theta$$

For each of the points given in polar coordinates, find two additional pairs of polar coordinates \((r, \theta),\) one with \(r>0\) and one with \(r<0\). $$\left(-2.7, \frac{5 \pi}{4}\right)$$

A gift shop sells figurines of famous people. Each figurine is mounted on a triangular base. The lengths of the edges of the base are \(4,5,\) and 6.5 inches. Find the area of the base.

Find the components of the vector in standard position that satisfy the given conditions. Magnitude \(10 ;\) direction \(190^{\circ}\)

Find the fourth roots of \(-8 i\)

In this set of exercises, you will use vectors and dot products to study real- world problems. In a new video game, Mario and Luigi are at positions defined by the vectors \langle 10,3\rangle and \(\langle x, 15\rangle .\) What must be the value of \(x\) so that their position vectors are orthogonal?

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

Find the components of the vector in standard position that satisfy the given conditions. Magnitude \(8 ;\) direction \(145^{\circ}\)

Find the sixth roots of 1

In this set of exercises, you will use vectors and dot products to study real- world problems. Design The position vectors of a tower and a small garden from the center of a fountain are given by \langle 50,60\rangle and \(\langle 40, y\rangle .\) Find \(y\) so that the two position vectors are orthogonal.

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

A billiard ball traverses a distance of 15 inches on a straight-line path, and then it collides with another ball, changes direction, and traverses a distance of 8 inches on a different straight-line path before coming to a stop. If the distance between the initial and final locations of the ball is 9 inches, find the measure of the angle formed by the lines that connect the initial location of the ball to the final location of the ball and to the point of the collision.