Engineering Practicum Baltimore Polytechnic Institute M. Scott Basic Engineering Math Objectives 1.Review basic Trig, Algebra, and Geometry 2.Understand and apply dimensional reasoning and the power-law expression 3.Apply basic finances 4.Perform error analysis on data and measurements

Engineering Practicum Baltimore Polytechnic Institute M. Scott 1.Total degrees in a triangle: 2.Three angles of the triangle below: 3.Three sides of the triangle below: 4.Pythagorean Theorem: x 2 + y 2 = r A B C x, y, and r y x r HYPOTENUSE A, B, and C Trigonometry

Engineering Practicum Baltimore Polytechnic Institute M. Scott Trigonometric functions are ratios of the lengths of the segments that make up angles.  y x r sin  = = opp. y hyp. r cos  = = adj. x hyp. r tan  = = opp. y adj. x Trigonometry

Engineering Practicum Baltimore Polytechnic Institute M. Scott a c A B C b Law of Cosines: c 2 = a 2 + b 2 – 2ab cos C Law of Sines: sin A sin B sin C a b c = Trigonometry

Engineering Practicum Baltimore Polytechnic Institute M. Scott 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 30 o. After walking 35 feet closer to the tree, the backpacker notes that the angle of elevation is 45 o. 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. 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 30 o 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? Engineering Problem Solving 1.Assign Variables, Write Given, Sketch and Label Diagram 2.Write Formulas / Equations 3.Substitute and Solve 4.Check Answer, THEN box answer

Engineering Practicum Baltimore Polytechnic Institute M. Scott 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 30 o 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? 200m 30 o Variables assigned x r tan 30 o = x / 200m x = (200m)tan 30 o x = (200m) * ( / 3) x = 115.5m cos 30 o = 200m / r r*cos 30 o = 200m r = 200m / cos 30 o r = 231m Equal signs aligned X: r:r:

Engineering Practicum Baltimore Polytechnic Institute M. Scott 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 Algebra

Engineering Practicum Baltimore Polytechnic Institute M. Scott 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 2.Multiplication in the Addition-or-Subtraction Method Algebra – Systems of Equations

Engineering Practicum Baltimore Polytechnic Institute M. Scott 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.) Notes

Engineering Practicum Baltimore Polytechnic Institute M. Scott Solve the following system by graphing: y = x 2 y = 8 – x 2 What is the solution? (2, 4) and (-2,4) Systems of Equations

Engineering Practicum Baltimore Polytechnic Institute M. Scott Solve the following system algebraically: 1) y = x 2 2) y = 8 – x 2 Substitute equation 1 into equation 2 and solve: x 2 = 8 – x 2 2x 2 = 8 x 2 = 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 Solution: (2, 4) and (-2,4) Systems of Equations

Engineering Practicum Baltimore Polytechnic Institute M. Scott Systems of equations can have: One Solution Multiple Solutions No Solutions Systems of Equations

Engineering Practicum Baltimore Polytechnic Institute M. Scott 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 units digit in the original numeral.

Engineering Practicum Baltimore Polytechnic Institute M. Scott Classwork: Systems of Equations – Word Problems 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.

Engineering Practicum Baltimore Polytechnic Institute M. Scott Area of a circle: pi*r 2 Volume of a sphere: (4/3)*pi*r 2 Volume of a cylinder: h*pi*r 2 Surface area of a sphere: 4*pi*r 3 Surface area of a cylinder: 2*pi*r 2 + 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)*A base *h Area of a circle: Volume of a sphere: Volume of a cylinder: Surface area of a sphere: Surface area of a cylinder: Surface area of a rectangular prism: Area of a triangle: Volume of a pyramid: Geometry Review

Engineering Practicum Baltimore Polytechnic Institute M. Scott 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. W. Wolfe

Engineering Practicum Baltimore Polytechnic Institute M. Scott 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 You have 5 minutes. DRILL

Engineering Practicum Baltimore Polytechnic Institute M. Scott Enrico Fermi, Physicist Fermi was one of the most notable physicists of the 20 th century. He is best known for his leading contributions in the Manhattan Project but his work spanned every field of physics. W. Wolfe

Engineering Practicum Baltimore Polytechnic Institute M. Scott 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.

Engineering Practicum Baltimore Polytechnic Institute M. Scott W. Wolfe 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. Laura and Enrico Fermi

Engineering Practicum Baltimore Polytechnic Institute M. Scott 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 W. Wolfe

Engineering Practicum Baltimore Polytechnic Institute M. Scott 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 W. Wolfe

Engineering Practicum Baltimore Polytechnic Institute M. Scott 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? W. Wolfe

Engineering Practicum Baltimore Polytechnic Institute M. Scott 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? W. Wolfe

Engineering Practicum Baltimore Polytechnic Institute M. Scott 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? W. Wolfe

Engineering Practicum Baltimore Polytechnic Institute M. Scott W. Wolfe 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.