Exercise 1.6: Explore how much conformal maps can distort size by visiting TheTrueSize.com website and answering the following questions. 1. Which is bigger Russia or the continent of Africa? 2. Which is bigger Poland or Kenya? 3. Which is bigger Alaska or Brazil?

The nth partial sum of a Fourier series is defined as: A_{0}+A_{1} \cos \omega_{1} t+B_{1} \sin \omega_{1} t+\ldots+A_{n} \cos \omega_{n} t+B_{n} \sin \omega_{n} t y(t)=\sum_{n=1}^{\infty} \frac{3 n}{2} \sin n+\frac{5 n}{3} \cos n t For the third partial sum of the Fourier series given by a) What is the fundamental frequency and the associated period? b) Express this partial sum as cosine terms only.

For a dynamic calibration involving a force measurement system, a known force is applied to a sensor. The force varies between 100 and 170 N at an angular frequency of @= 10 rad/s.Find the average (static) value of the input signal, its amplitude and its frequency in Hertz.Assuming that the signal may be represented by a simple periodic waveform, express the signal, y(t), as a one term Fourier series. For 10 bonus points create an amplitude spectrum from an equivalent discrete time data set.

Consider the triangle wave shown in the Figure below as a periodic function with period 2n. Determine the Fourier series that represents this function, y(t). Plot the first three partial sums over the range from -n to +n.

Determine the average and RMS values (ỹ and Yrms) for the function: y(t)=25+10 \sin 6 \pi t Over the time periods t1 to t2 listed below а) 0 to 0.1 sec b) 0.4 to 0.5 sec c) 0 to 1/3 sec d) 0 to 20 sec Comment on the nature and meaning of the results in terms of analysis of dynamic signals.

Sketch representative waveforms of the following signals: a) thermostat on a refrigerator b) input to a spark plug c) input to a cruise control d) a pure musical tone e) note produced by a guitar string f) AM and FM radio signals

3. (8 pts) Prove the following statement:If A, B, and C are points with A * B * C, then AB = AČ.

1. Consider an interpretation where "points" are the numbers 1,2,3 and 4, "lines" are the sets {1,2}, {1,3}, {1,4}, and {2,3, 4}, and “lies on" means "is contained in." (b) (6 pts) Which (if any) parallel postulate does this interpretation satisfy? Prove your answer. (a) (8 pts) Is this a model for incidence geometry? Prove your answer. (c) (4 pts) Is this a model for neutral geometry? Prove your answer.

5. Which of the datum features represents a.an axis or median plane b. a surface

Find the air-fuel ratio of a 4-stroke, 1 cylinder, air cooled engine with fuel consumption time for 10 cc as 20.0 sec. and air consumption time for 0.1 m³ as 16.3 sec. The load is 16 kg at speed of 3000 rpm. Also find brake specific fuel consumption in g/kWh and thermal brake efficiency. Assume the density of air as 1.175 kg/m³ and specific gravity of fuel to be 0.7. The lower heating value of fuel is 44 MJ/kg and the dynamometer constant is 5000.