EPS 5 Problem Set #0 Due: Wed. Feb. 3rd at beginning of lecture ...

EPS 5
Problem Set #0
Due: Wed. Feb. 3 rd at beginning of lecture
This problem set reviews some of the basic mathematical concepts we will be using throughout
the semester in EPS 5. The problems presented here are representative of skills you may need to
solve problems in the course, but are not exhaustive. You should be comfortable with math
through algebra, geometry, and pre-calculus. Calculus is not used in EPS 5. You may use a
calculator to do the following problems. Where appropriate, show your work for partial credit.
1. Scientific notation (2 points)
Scientific notation allows us to represent numbers with a convenient shorthand notation by
expressing them as powers of ten:
a × 10 b
where the exponent (b) is an integer and the base or coefficient (a) is a real number usually greater
than or equal to 1 and less than 10. For example, one hundred twenty three billion in scientific
notation can be written as:
123,000,000,000 = 1.23×10 11 = 1.23E+11
Write the following in scientific notation:
a) 0.0000000102 = 1.02 x 10 -8
b) 54330070000 = 5.433007 x 10 10
2. Significant Figures (2 points):
“Significant figures” roughly represents the error in a quantity by the number of digits recorded,
revealing how precisely that quantity is known. The general rule for significant figures is to include
only as many digits as you know for the values used in the calculation.
• Non-zero digits are always significant;
• Any zeros between two significant digits are significant;
• Zeros that occur at the end of a number on the right-hand side of a decimal point are significant.
For example, 6.302×10 -6 has four significant figures. The same number could be expressed as
0.000006302, but it would still have only four significant figures. The leading zeros are not
significant and are only stating the number of decimal places.
Example: 11.1/3.2 = 3.5
Solve the following:
a) What is the square root of 18.24? = 4.271
b) What is 54.2 divided by 2.5 ? = 22

3. Exponents, Exponentials, and Logarithms (3 points):
“log” refers to a logarithm with a base of 10, while “ln” is a “natural logarithm” and has a base of e
(e = 2.71828...). All of the rules work for either log or ln, but only the ln form is presented here.
Solve the following expressions:
a) (10 -3 )×(2.9×10 10 ) = 2.9 x 10 7
b) log10(10 -5 ) = -5
c) e ln(18) = 18
4. Degrees and Radians (3 points):
ln(e x ) = e ln(x) = x
ln(x a ) = a ln(x)
ln(x)-ln(y) = ln(x/y)
ln(x) + ln(y) = ln(xy)
(x a ) (x b ) = x a+b
(x a ) b = xa b
x 1/2 = square root of x
x -a = 1/x a
x 0 = 1
In science (particularly planetary science) we often deal with angles. We usually use degrees (360 o
for full circle), but may use radians. 2π radians equal 360 o . Be sure to set your calculator to
DEGREE MODE if you are using degrees and RADIAN MODE if you are using radians. (Review:
http://www.clarku.edu/~djoyce/trig/angle.html)
Solve the following questions (2 significant figures):
a) cos(π/3) = 0.50
b) sin(45 o ) = 0.71
c) tan(3π/4) = -1.0
5. Solving for variables (2 points):
In this class we will often need to rearrange formulas to solve for a quantity of interest.
Solve the following:
a) (3x + 4y) 1/2 = (12/y) ; Solve for x in terms of y. x = (48/y 2 ) - (4/3)y
b) x = ln(3y + 2) ; Solve for y in terms of x. y = (1/3)(e x - 2)

6. Unit conversions (5 points):
Keeping track of your units is VERY important and will aid you in problem solving!
Convert the following:
a) How many m 3 in km 3 ? = 1 x 10 9
b) If an object has a velocity of 100 km/h, what is its velocity in cm/s? 2.78 x 10 3
c) The area of my property is 21.3 hectares. What is this in km 2 ? 0.213
d) The atmospheric pressure in Cambridge today is 0.981 atm. What is the pressure in pascal?
9.94 x 10 4
e) The temperature in Cambridge today is 6 o C. What is the temperature in o F? 43 o
7. Area of continents and sea level rise (5 points):
The average radius of Earth is 6.38×10 6 m and the combined volume of ice in the Antarctic and
Greenland ice caps is 30.2×10 6 km 3 . Continents cover 29.2% of Earth’s surface.
Solve the following:
a) What is the surface area of the earth in km 2 ? 5.12 x 10 8
b) What is the total area of the continents in km 2 ? 1.49 x 10 8
c) What is the rise in sea level in meters if all of the ice in Greenland and the Antarctic were to melt?
Assume the total area of continents does not change. (Note: a given mass of water occupies 92.0%
the volume of the same mass of ice) (3 points) 76.8 m