Transcript Basic Engineering Math - Baltimore Polytechnic Institute

Basic Engineering Math
1.
2.
3.
4.
Objectives
Review basic Trig, Algebra, and Geometry
Understand and apply dimensional
reasoning and the power-law expression
Apply basic finances
Perform error analysis on data and
measurements
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Trigonometry
1.
2.
3.
4.
Total degrees in a triangle: 180
Three angles of the triangle below: A, B, and C
Three sides of the triangle below: x, y, and r
B
Pythagorean Theorem:
x2 + y2 = r2
r
A
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y
x
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C
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Trigonometry
Trigonometric functions are ratios of the lengths of the
segments that make up angles.
opp.
sin Q =
hyp.
y
=
r
adj.
cos Q =
hyp.
x
=
r
opp.
tan Q =
adj.
y
=
x
r
y
Q
x
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Trigonometry
B
Law of Cosines:
c2 = a2 + b2 – 2ab cos C
c
a
A
b
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C
Law of Sines:
sin A
sin B
sin C
=
=
a
b
c
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A: A radio antenna tower stands 200 meters tall. A supporting cable
attached to the top of the tower stretches to the ground and makes a
30o angle with the tower. How far is it from the base of the tower to
the cable on the ground? How long must the cable be?
B: A backpacker notes that from a certain point on level ground, the
angle of elevation to a point at the top of a tree is 30o. After walking
35 feet closer to the tree, the backpacker notes that the angle of
elevation is 45o. The backpacker wants to know the height of the tree.
Draw and label a sketch of the backpacker’s situation and create two
equations based on the two triangles of your sketch.
Engineering Problem Solving
1. Assign Variables, Write Given, Sketch and Label Diagram
2. Write Formulas / Equations
3. Substitute and Solve
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4.
Check Answer, THEN box
answer
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DRILL A: RADIO TOWER – SOLUTION
A radio antenna tower stands 200 meters tall. A supporting cable
attached to the top of the tower stretches to the ground and makes a
30o angle with the tower. How far is it from the base of the tower to
the cable on the ground? How long must the cable be?
tan 30oassigned
= x / 200m
X:Variables
x = (200m)tan 30o
30o
x = (200m) * ( 3 / 3)
200m
x = 115.5m
r
r:
cos 30o = 200m / r
r*cos 30o = 200m
r = 200m / cos 30o
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x
Equal signs aligned
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r = 231m
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Algebra
Read the entire problem through. Note that not all information given is
relevant.
1. Assign Variables, Write Given, Sketch and Label Diagram
1. Whenever you write a variable, you must write what that variable
means.
2. What are the quantities? Assign variable(s) to quantities.
3. If possible, write all quantities in terms of the same variable.
2. Write Formulas / Equations
What are the relationships between quantities?
3. Substitute and Solve
Communication: All of your work should communicate your thought
process (logic/reasoning).
4. Check Answer, then Box Answer
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Algebra – Systems of Equations
•
Because two equations impose two conditions on the
variables at the same time, they are called a system of
simultaneous equations.
• When you are solving a system of equations, you are
looking for the values that are solutions for all of the
system’s equations.
• Methods of Solving:
1. Graphing
2. Algebra:
1. Substitution
2. Elimination
1. Addition-or-Subtraction
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2. Multiplication in the Addition-or-Subtraction Method
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Notes
•
Systems of Equations:
– Use the multiplication / addition-or-subtraction
method to simplify and/or solve systems of
equations:
• Eliminate one variable by adding or
subtracting corresponding members of the
given equations (use multiplication if
necessary to obtain coefficients of equal
absolute values.)
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Systems of Equations
•
Solve the following system by graphing:
y = x2
y = 8 – x2
What is the solution?
(2, 4) and (-2,4)
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Systems of Equations
•
Solve the following system algebraically:
1) y = x2
2) y = 8 – x2
Substitute equation 1 into equation 2 and solve:
x2 = 8 – x2
2x2 = 8
x2 = 4
x = 2 and -2
Now substitute x-values into equation 1 to get y-values:
when
x = 2,
y=4
when
x = -2, y = 4
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Solution: (2, 4) and (-2,4)
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Systems of Equations
•
Systems of equations can have:
One Solution
Multiple Solutions
No Solutions
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Examples:
Algebra A:
Jenny and Kenny together have 37 marbles, and Kenny has
15. How many does Jenny have? (Solve algebraically, then
graphically to check.)
Algebra B:
The admission fee at a small fair is $1.50 for children and $4.00 for
adults. On a certain day, 2200 people enter the fair and $5050 is
collected. How many children and how many adults attended?
Algebra C:
The sum of the digits in a two-digit numeral is 10. The number
represented when the digits are reversed is 16 times the original
tens digit. Find the original two-digit number.
Hint: Let t = the tens digit in the original numeral and u = the
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units digit in the original numeral.
Systems of Equations – Word Problems
Classwork:
Algebra A:
The perimeter of a rectangle is 54 centimeters. Two times the
altitude is 3 centimeters more than the base. What is the area
of the rectangle?
Algebra B:
Three times the width of a certain rectangle exceeds twice its
length by three inches, and four times its length is twelve more
than its perimeter. Find the dimensions of the rectangle.
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Geometry Review
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•
•
Area of a circle: pi*r2
Volume of a sphere: (4/3)*pi*r2
Volume of a cylinder: h*pi*r2
Surface area of a sphere: 4*pi*r3
Surface area of a cylinder: 2*pi*r2 + 2*pi*r*h
Surface area of a rectangular prism: 2*a*b + 2*a*c + 2*b*c
Area of a triangle: (1/2)*b*h
Volume of a pyramid: (1/3)*Abase*h
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Fermi Problem Example
“What is the length of the equator?”
Fermi problems are solved by assembling simple facts that
combine to give the answer:
•The distance from Los Angeles to New York is about 3000 miles.
• These cities are three time zones apart.
• So each time zone is about 1000 miles wide.
• There are 24 time zones around the world.
• So the length of the equator must be about 24,000 miles
The exact answer is 24,901 miles.
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DRILL
Problem:
In your group of 3, estimate
the volume of water, in liters,
of the world’s oceans. You
must list (could be one list):
1) all assumptions
2) the logic of your thinking
3) conversions
Work should be neat and
easily followed.
Calculators are not allowed.
You may only use the picture
to the right and any prior
knowledge
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5 minutes.
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Enrico Fermi, Physicist
Fermi was one of the most
notable physicists of the
20th century.
He is best known for his
leading contributions in
the Manhattan Project
but his work spanned
every field of physics.
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Fermi Electron Theory
While in Pisa, Fermi and his
friends had a well-earned
reputation as pranksters.
One afternoon, while patiently
trapping geckos (used to scare
girls at university), Fermi came
up with the fundamental theory
for electrons in solids.
Fermi’s theory later became the
foundation of the entire
semiconductor industry.
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Fermi Problems
Fermi was famous for being able to avoid long, tedious
calculations or difficult experimental measurements
by devising ingenious ways of finding approximate
answers.
He also enjoyed challenging his
friends with “Fermi Problems”
that could be solved by such
“back of the envelope”
estimates.
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Laura and Enrico
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W. Fermi
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Fermi Problems
• Open ended problem solving.
• Thought process is more important than calculating
exact answer.
• Steps in solving Fermi problems
– Determine what factors are important in solving
problem
– Estimate these factors
– Use dimensional reasoning to calculate a solution
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Fermi Problems
• What do Fermi Problems have to do with
engineering
– Engineers have to solve open ended problems that
might not have a single right solution
– Engineers have to estimate a solution to a
complicated problem
– Engineers have to think creatively
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Ping Pong Anyone?
Solving a problem in 60 seconds (individually)
Look around the room you are sitting in. Take just 60 seconds to
answer the following questions:
How many ping-pong balls could you fit into the room?
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Ping Pong Anyone?
Solving a problem in 60 seconds (individually)
Look around the room you are sitting in. Take just 60 seconds to
answer the following questions:
How many ping-pong balls could you fit into the room?
What was your model of a ping-pong ball? What was your
model of the room?
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Ping Pong Anyone?
Solving a problem in 60 seconds (individually)
Look around the room you are sitting in. Take just 60 seconds to
answer the following questions:
How many ping-pong balls could you fit into the room?
What was your model of a ping-pong ball? What was your
model of the room?
What other simplifications or assumptions did you make?
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The Fermi Paradox
The extreme age of the universe and its vast number of
stars suggest that if the Earth is typical,
extraterrestrial life should be common.
Discussing this proposition with colleagues over lunch
in 1950, Fermi asked: "Where is everybody?”
We still don’t have a
good answer to Enrico’s
question.
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