A221 Mathematics Review Questions


Nota Bene: If you had difficulties with any of these exercises, please see Prof. van Zee or Ms. Maria Cordero for assistance.

Scientific Notation

1. Express the following in scientific notation:
a) 24.16827 = 2.416827 x 101
b) 71283.6 = 7.12836 x 104
c) 5300000 = 5.3 x 106
d) 0.0005836 = 5.836 x 10-4
e) 5.5 x 101 x 2.23 x 105 = 1.2 x 107
f) 2.18 x 102 x 3. x 10-4 = 6. x 10-2
g) 6.24 x 102 + 3. x 102 = 9. x 102

2. A rectangular plate has a length of 25.3 +/- 0.5 cm and a width of 5.75 +/- 0.06 cm. Find the area of the plate and the uncertainty in the final result.

A = 25.3 x 5.75 cm2 = 145. cm2
error = +/- area * sqrt((0.5/25.3)**2 + (0.06/5.75)**2) = +/- 3. cm2

Powers, Roots, and Logs

3. Evaluate the following without using a calculator:
a) 53 = 125
b) 160.5 = 4
c) 9-0.5 = 1/3
d) log(100) = 2
e) log(0.01) = -2

Trigonometry

Right Triangle
4. Express the following in terms of a, b, and c:
a) sin A = a/c
b) cos A = b/c
c) tan A = a/b

5. If a=3.0 cm and b=4.0 cm, calculate the length of c.
c2 = a2 + b2
c2 = 9 + 16 = 25
c = 5

6. A radian is defined as the angle, Theta, when the arclength,s, of the inscribed angle is equal to the radius r. Convert the following angular measures into radians:

Circle

a) 60o = pi / 3 = 1.047
b) 180o = pi = 3.1416
c) 45o = pi / 4 = 0.7854

Algebra

7. Solve for x:
a) a= 1 / (1 + x) : x = (1 - a)/a
b) 3x - 5 = 13 : x = 6
c) ax - 5 = bx + 2 : x = 7 / (a - b)

Unit Conversions

We will use standard SI units throughout most of this class, but there are a few astronomically convenient units that will also be used, when appropriate.
1 AU = 1.496 x 1011 m
1 pc = 2.06265 x 105 AU
1 ly = 6.324 x 104 AU
1 Msun = 1.99 x 1030 kg

8. Convert the following distances into standard SI units:

a) 5.203 AU = 7.784 x 1011 m
b) 39.44 AU = 5.900 x 1012 m
c) 2.702 x 105 AU = 4.042 x 1016 m
d) 1.31 pc = 4.04 x 1016 m
e) 65 Mpc = 2.0 x 1024 m

Vectors

See http://www.physics.uoguelph.ca/tutorials/vectors/vectors.html for a review of vector addition.