Lab1 - Worksheet (1)
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University of California, Berkeley *
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102
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Feb 20, 2024
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ENERES 102 | Lab 1 – Estimation Introduction
A key component of this course is learning how to break down complex real life phenomena into simplified scientific models. Our textbook, Consider a Spherical Cow (Harte), is named after a joke amongst theoreticians in which such a framework is applied to assist a dairy farmer in increasing milk production through simplifying each cow as a sphere. The main requirement of such an approach is to strip down a problem to just the essentials, removing all unnecessary detail. Developing a good sense of how to use your physical intuition for solving problems involves learning the technique of deciding which approach to solving a problem tends to work more "easily" than other possible approaches. In some cases, detailed mathematical analysis is necessary. However, for many questions, a surprisingly high degree of accuracy can be obtained by a series of "educated guesses". Nuclear physicist, Enrico Fermi, is acknowledged as one of the great masters of this latter approach. Hence, the kind of problems that can be approached in this manner are termed
Fermi problems
, also referred to as Back-Of-The-Envelope Calculations. A general approach to this is as follows:
Break up the problem
. If you can’t estimate an answer, break the problem into smaller pieces and estimate the answer for each one.
Estimate by bounding
. Sometimes it’s easier to start with setting lower and upper bounds.
Observe the world around you
. Think about what you do know.
Use common sense to check your answers
. Ask yourself, does this seem reasonable?
Some more general tips regarding the math in Back of the Envelope Calculations:
●
Use significant figures. Never report more precision than you think is justified in a calculation.
○
Adding/subtracting: round the answer to the least number of decimal places.
○
Multiplying/dividing: round the answer to the least number of significant digits.
●
Use scientific notation. It helps report the correct number of significant figures. If you have fewer
sig figs than digits before the decimal point, use scientific notation. Much of this course is estimation, so we want to see simpler numbers in scientific notation than unreasonably precise answers with lots of digits.
●
Unit conversion. Always multiply by one (e.g. to switch from meters to kilometers, multiple by 1km/1000m) and carry the units.
●
Rule of thumb for guessing or looking up figures - if it’s in Harte (the course textbook) use it. If it’s not, use your best guess.
●
Memorize key unit conversions (
Harte appendix) and report answers in SI units unless requested otherwise
Problems
Part 1. Let’s practice this with the following problems:
1. What is the number of new passenger cars sold each year in the USA?
2. Fans of a college football team are excited after their team wins the game. They rush onto the field. How many people can fit onto the field of 100 yards by 50 yards?
Part 2. Basic Math Warm-Up
These next set of problems review basic mathematical methods frequently used in environmental science and energy analysis. They provide an opportunity for you to brush up on math skills and identify areas where further work will be needed to come up to speed. Do all the problems “by hand” – no calculator is required for any problem.
2.1 Scientific notation
6. Express the following numbers in scientific notation:
a. 209,652,201
b. –756
c. 47,559.8
d. 0.00386
7. Write out the following as ordinary decimal numbers:
a. 3.15 x 10
7
b. 1.8678 x 10
-3
c. 5.9 x 10
-5
8. Calculate the following:
a. 10
3
x 10
5
=
b. 10
-2
x 10
14
=
c. 10
3
÷ 10
8
=
d. 10
-6
÷ 10
-9
=
e. 10
2
+ 10
1
=
9. Calculate the following and express the answer with the correct number of significant figures:
a. (1.6 x 10
5
) x (3.000 x 10
-3
) =
b. (3.6 x 10
-3
) ÷ (1.20 x 10
-6
) =
c. (2.705 x 10
4
) + (2.95 x 10
4
) =
d. (1.28 x 10
-3
) – (7.8 x 10
-4
) =
2.2 Metric prefixes and units
10. Give (i) the full name and (ii) the number (in scientific notation) that is the equivalent of each of the metric prefix abbreviations below:
a. c
b. M
c. G
d. µ
e. E
11. Give the metric prefix (both the full name and the one-letter abbreviation) that is the equivalent of the number below:
a. 10
3
b. 10
-3
c. 10
12
d. 10
-9
e. 10
15
12. Write the SI abbreviation for each of the following units (e.g. megawatt = MW):
a. kilogram
b. millimeter
c. exajoule
d. microsecond
13. Convert the following into the form prefix + unit (e.g. 3 x 1012 g = 3 Tg) :
a. 7 x 10
-6
s
b. 1.5 x 10
13
W
c. 92,000 g
d. 4.5 x 10
-7
m
2.3 Exponential and logarithmic functions
14. Solve the following
a. 10
x
• 10
-x
=
b. (10
x
)
y
=
c. 2
3
÷ 2
5
=
d. e
r
• e
s
÷ e
t
=
e. 10
at
• 10
-bt
=
15. Solve the following. (Note: “log” written without a subscript is shorthand for log
10
)
a. log (ab) =
b. log (a/b) =
c. log a
b
=
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g. log
e
(ab) =
f. ln e
-x
=
16. Solve the following without using a calculator. (Hint: log 2 ≈ 0.30, log 3 ≈ 0.48, ln 2 ≈ 0.69, ln 10 ≈ 2.3)
a. log
3
9 =
b. log
9
3 =
c. log 4 =
d. log 500 =
e. log 30,000 =
h. ln 100 =
i. ln 0.5 =
17. What is the doubling time (or halving time) if the exponential growth (or decay) rate is the following: (hint: exponential growth: P = P
o
e
rt
)
a. 4.6% per year
b. 0.23/s c. 0.0069/min
18. A rabbit population grows exponentially at a rate of 10% per month. (a) In about how many months will the population double? (b) In about how many months will the population have doubled 20 times? (c) About how many times the original population will the rabbit population be
after 20 doublings? (Hint: Use the fact that 2
10
= 1024 ≈ 10
3
to calculate the population).
19. Make a data table for the functions y = 2
x
and y = 10
x
for integer values of -3 ≤ x ≤ 3. 2.4 Geometrical areas and volumes
20. Calculate the following:
a. The area of a square 6 feet on a side.
b. The (i) circumference and (ii) area of a circle with a radius of 5 miles.
c. The area of a triangle with a height of 20 cm and a base of 10 cm.
d. The (i) surface area and (ii) volume of an ice cube 1 inch on each edge.
e. The (i) radius and (ii) volume of a sphere with a circumference of 62.8 m.
2.5 Trigonometry
21. Convert the following angles from degrees to radians and vice versa:
a. 360º
b. 45º
c. 2π/3
d. - π/2
22. For the right triangle below, write the formulas for sin A, cos A, and tan A in terms of the
lengths of the sides x, y, and z.
23. Write the formula for tan A in terms of sin A and cos A.
24. (a) If angle A equals 30º and side z has a length of 10 miles, what are the lengths of sides x and y, respectively? [Hint: cos 30º = 0.866, sin 30º = 0.5).] (b) Check your answer using the Pythagorean theorem.
Part 3. Excel
Excel Basics & Useful Functions:
= SUM( )
= MAX( )
= MIN( )
= EXP( )
= A1+A2
= A1*A2
= A1/A2
= $A$1 absolute reference
= IF(logical)
1. Using Excel, graph y = 0.5sin(x+2)+7 on the interval [-10, 10] 2. The population in a town was 3,810 in 2007 and is growing at an annual rate of 3.5%. If this growth rate continues, what will be the approximate population in 2020? 2050? Graph the population in excel. P = P
o
e
rt
3. Make a data table showing the values of x and y for the following functions, for each integer value of x from -3 to +3. (Hint: if necessary, start by algebraically manipulating the equation into a form where you can easily find the value of y).
a. y = 2x
b. –2x = 4y + 4
c. y – 5 = x
2
For the functions above: (d) what are the slopes and y-intercepts of equations (a) and (b)? (e) what type of function is equation (c)?
Brushing Up Resources:
Pre-Algebra, Exponents, & Radicals: https://www.khanacademy.org/math/pre-algebra/pre-algebra-exponents-radicals
Appropriate Units in Modeling: https://www.khanacademy.org/math/algebra-home/alg-
modeling/alg-appropriate-units
Exponential Growth vs. Linear growth : https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:exponential-growth-
decay/x2f8bb11595b61c86:exponential-vs-linear-growth-over-time/v/exponential-vs-
linear-growth-over-time
Exponential and Logarithmic Functions: https://www.khanacademy.org/math/algebra-
home/alg-exp-and-log
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