Download Review for Exam 4

Math 2412 Review Items for Exam 4
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Solve the triangle.
1)
Solve the problem.
10) Two airplanes leave an airport at the same
time, one going northwest (bearing 135°) at 409
mph and the other going east at 325 mph. How
far apart are the planes after 4 hours (to the
nearest mile)?
7
15°
115°
Use Heron's formula to find the area of the triangle.
Round to the nearest square unit.
11) a = 9 inches, b = 15 inches, c = 7 inches
Solve the triangle. Round lengths to the nearest tenth and
angle measures to the nearest degree.
2) A = 26°, B = 51°, c = 24
Plot each point in polar coordinates on the graph.
π
3π
3π
12) A (4, 0), B (-4, - ), C (4, ), D (-4,
)
2
4
4
Two sides and an angle (SSA) of a triangle are given.
Determine whether the given measurements produce one
triangle, two triangles, or no triangle at all. Solve each
triangle that results. Round lengths to the nearest tenth
and angle measures to the nearest degree.
3) A = 30°, a = 22, b = 44
5
4) B = 31°, b = 14, a = 26
-5
5
5) C = 35°, a = 18.7, c = 16.1
Find the area of the triangle having the given
measurements. Round to the nearest square unit.
6) A = 30°, b = 15 inches, c = 5 inches
-5
Find another representation, (r, θ), for the point under the
given conditions.
π
13) 1,
, r < 0 and 0 < θ < 2π
4
Solve the problem.
7) Two tracking stations are on the equator 162
miles apart. A weather balloon is located on a
bearing of N 43°E from the western station and
on a bearing of N 22°E from the eastern station.
How far is the balloon from the western
station? Round to the nearest mile.
Polar coordinates of a point are given. Find the rectangular
coordinates of the point.
2π
14) 9,
3
Solve the triangle. Round lengths to the nearest tenth and
angle measures to the nearest degree.
8)
15) (5, 41°)
The rectangular coordinates of a point are given. Find
polar coordinates of the point.
16) (6 3, 6)
9
6
4
Convert the rectangular equation to a polar equation that
expresses r in terms of θ.
17) y = 3
9) b = 7, c = 8, A = 109°
1
Plot the complex number.
26) A: - 3 + 6i, B: 2i, C: -2 - i, D: 6 2 - 6 2i
18) (x - 3)2 + y2 = 9
i
Convert the polar equation to a rectangular equation.
19) r = 8
10
5
20) r = 8 cos θ + 4 sin θ
-10
Test the equation for symmetry with respect to the given
axis, line, or pole.
21) r = 2 cos θ; the polar axis
-5
5
10
R
-5
-10
22) r = 2 + 2 cos θ; the line θ =
π
2
Find the absolute value of the complex number.
27) z = -14 - 8i
π
23) r = 6 + 2 sin θ; the line θ =
2
Write the complex number in polar form. Express the
argument in degrees.
28) 2i
Graph the polar equation.
24) r = 6 sin θ
6
5
4
3
2
1
-6 -5 -4 -3 -2 -1-1
-2
-3
-4
-5
-6
Write the complex number in polar form. Express the
argument in radians.
29) - 2 3 - 2i
Write the complex number in rectangular form.
2π
2π
30) 3(cos
)
+ i sin
3
3
1 2 3 4 5 6 r
31) 6(cos π + i sin π)
Find the product of the complex numbers. Leave answer in
polar form.
32) z 1 = 5(cos 34° + i sin 34°)
25) r = 3 sin 2θ
6
5
4
3
2
1
-6 -5 -4 -3 -2 -1-1
-2
-3
-4
-5
-6
z 2 = 3(cos 18° + i sin 18°)
Find the quotient
1 2 3 4 5 6 r
z1
z2
of the complex numbers. Leave
answer in polar form.
33) z 1 = 5(cos 200° + i sin 200°)
z 2 = 4(cos 50° + i sin 50°)
Use DeMoivre's Theorem to find the indicated power of
the complex number. Write answer in rectangular form.
3π
3π 3
34) 10 (cos
)
+ i sin
4
4
2
Use the dot product to determine whether the vectors are
parallel, orthogonal, or neither.
49) v = 3i + j, w = i - 3j
35) (1 - i)10
Find all the complex roots. Write the answer in the
indicated form.
36) The complex fourth roots of -16 (rectangular
form)
Find projwv.
50) v = 2i + 3j; w = 8i - 6j
Find the magnitude v of the vector.
37) v = 9i + 12j
Decompose v into two vectors v1 and v2 , where v1 is
parallel to w and v2 is orthogonal to w.
51) v = -3i + 5j, w = 2i + j
Draw a sketch to represent the vector. Refer to the vectors
pictured here.
Solve the problem.
52) A force is given by the vector F = 5i + 7j. The
force moves an object along a straight line
from the point (10, 6) to the point (12, 19). Find
the work done if the distance is measured in
feet and the force is measured in pounds.
38) 3d
39) -b
40) b - c
A vector v has initial point P1 and terminal point P2 . Write
v in terms of ai + bj.
41) P1 = (-2, 3); P2 = (-5, -6)
Find the specified vector or scalar.
42) u = -4i - 2j, v = 6i + 7j; Find u - v.
43) v = 3i + 4j; Find 2v.
44) u = -8i + 1j and v = 6i + 1j; Find u + v .
Find the unit vector having the same direction as v.
45) v = 12i + 5j
Write a vector v in terms of i and j whose magnitude v
and direction angle θ are given.
46) v = 8, θ = 30°
Use the given vectors to find the specified scalar.
47) u = -8i + 5j and v = -11i - 6j; Find u · v.
Find the angle between the given vectors.
48) u = i - j, v = 4i + 5j
3
Answer Key
Testname: 2412REVIEW4SP2011
1)
2)
3)
4)
5)
C = 50°, a = 24.51, c = 20.72
C = 103°, a = 10.8, b = 19.1
B = 90°, C = 60°, c = 38.1
no triangle
A1 = 42°, B1 = 103°, b1 = 27.4;
24)
6
5
4
3
2
1
A2 = 138°, B2 = 7°, b2 = 3.4
6) 19 square inches
7) 419 miles
8) A = 127.2°, B = 32.1°, C = 20.7°
9) a = 12.2, B = 33°, C = 38°
10) 2716 miles
11) 22 square inches
5
25)
B
6
5
4
3
2
1
A
-5
5
D
-5
12)
5
π
4
26)
i
9 9 3
14) - ,
2 2
10
A
5
15) (3.774, 3.28)
π
16) 12,
6
B
-10
17) r sin θ = 3
18) r = 6 cos θ
19) x2 + y2 = 64
-5
5
10
-5
D
20) x2 + y2 = 8x + 4y
21) has symmetry with respect to polar axis
22) may or may not have symmetry with respect to the
π
line θ =
2
23) has symmetry with respect to the line θ =
1 2 3 4 5 6 r
-6 -5 -4 -3 -2 -1-1
-2
-3
-4
-5
-6
C
13) -1,
1 2 3 4 5 6 r
-6 -5 -4 -3 -2 -1-1
-2
-3
-4
-5
-6
-10
27) 2 65
28) 2(cos 90° + i sin 90°)
7π
7π
29) 4 cos
+ i sin
6
6
π
2
30) -
3 3 3
i
+
2
2
31) -6
32) 15(cos 52° + i sin 52°)
5
33) (cos 150° + i sin 150°)
4
4
R
Answer Key
Testname: 2412REVIEW4SP2011
34) 500 2 + 500 2i
35) -32i
36) 2 + 2i, 2 - 2i, 37) 15
38)
2+
2i, -
2-
2i
39)
40)
41) v = -3i - 9j
42) -10i - 9j
43) 6i + 8j
44) 2 2
12
5
45) u =
i+
j
13
13
46) v = 4 3i + 4j
47) 58
48) 96.3°
49) orthogonal
-1
50)
(4i + 3j)
25
51) v1 = -
1
13
26
(2i + j), v2 = i+
j
5
5
5
52) 101 foot-pounds
5