ISE 362 HOMEWORK SIX Due Date: Thursday 11/30

1. A cone and plate viscometer (shown below) is a torque meter which is driven at discrete rotational speeds. The torque measuring system, which consists of a calibrated spring connecting the drive mechanism to a rotating cone, senses the resistance to rotation caused by the presence of a fluid between the cone and a stationary flat plate. The resistance to the rotation of the cone produces a torque that is proportional to the shear stress in the fluid. In an experiment, the viscosity of a new motor oil was measured by a cone and plate viscometer at six different cones speeds. It was decided that a quadratic regression model was appropriate, and the estimated regression function resulting from the n = 6 observations was: y = –0.0178x2 + 3.3684x –113.0937 2 2 Where Σyi = 210.7, Σy i = 8,386.43, Σxiyi = 17,002.0 and Σx i yi = 1,419,780.0. (Viscosity readings are in absolute centipoise units >> mPa). Find the values listed below: SSE =

What viscosity would you predict for a cone speed of 60 rpm? Yx=60 =

2. Chemical preservatives are frequently used in processed foods to prevent growth of bacteria, yeast or other microorganisms. Sodium Benzoate (NaC6H5CO2) is a type of preservative commonly used in fruit pies, jams, beverages, salads, relishes and sauerkraut. Typically, foods that have an acidic pH because there is a marked pH effect for Sodium Benzoate: the lower the pH, the more effective it is. An investigation of the influence of Sodium Benzoate concentration on the critical minimum pH necessary for inhibiting the growth and activity of an iron and sulphur-oxidizing bacterium, Thiobacillus Ferrooxidians, yielded the accompanying data, which suggests that expected critical minimum pH is linearly related to the natural logarithm of molar concentrate: Molar Concentration >> 0.01 0.025 0.1 0.95 pH >> 5.1 5.5 6.1 7.3 What is the implied probabilistic model, and what are the estimates of the model parameters? Also, what critical minimum pH would you predict for a molar concentration of 1.0? Ans:

3. During the original creation of the ISE intramural softball team, a young student engineer from the freshman class, named Dan, tried out for the infield position at third base. Dan had a keen eye for hitting, above average fielding range, a vacuum cleaner for a glove, and an exceptional rocket arm to deliver the ball to first base. Unfortunately, there was one major fault; Dan could never throw out anyone at first base. His natural release would always make the ball curve away from the first baseman. With help from his ISE classmates, Dan discovered in lab that as the ball moves through the air toward first base, its velocity creates an air stream moving against the trajectory of the ball itself (see drawing on next page). Imagine it as two lines, one curving over the ball and one curving under, as the ball moves in the opposite direction. In an ordinary throw, the effects of the airflow would not be particularly intriguing, but in this case, Dan’s release naturally placed a "spin" on the ball. If the direction of airflow is from right to left, the ball, as it moves into the airflow, is spinning clockwise. This means that the air flowing over the ball is moving in a direction opposite to the spin, whereas that flowing under it is moving in the same direction. The opposite forces produce a drag on the top of the ball, and this cuts down on the velocity at the top compared to that at the bottom of the ball, where spin and airflow are moving in the same direction. Thus, the air pressure is higher at the top of the ball making the ball curve toward the side of lower air pressure. The results of the lab experiment are listed below with a demonstrative graph illustrating lower air pressure versus the amount of curve in degrees. Dan needs a model so he can predict how much the ball will curve toward the side of lower air pressure. Estimate the parameters of a regression function and the regression function itself. BTW: Dan Bernoulli never became a great softball player; however, he did become a great mathematician. Make sure your regression model provides a Coefficient of Determination greater than R2>0.98.

Experimental Data (12 Data points): Lower Air Pressure (Atm) Deflection (Degrees) L_Air Pressure (Atm) Deflection (Degrees) .05 > 118.8 .60 > 9.8 .10 > 61.5 .70 > 8.6 .20 > 29.2 .80 > 8.0 .30 > 19.9 .90 > 6.7 .40 > 14.8 1.0 > 6.3 .50 > 12.5 1.1 > 5.5 P l o t o f C u r v e E x p e r i m e n t 1 4 0

) 1 0 0 s e e r g

e 8 0 D (

n o i 6 0 t c e l f e

0 0 0 . 2 0 . 4 0 . 6 0 . 8 1 1 . 2 1 . 4 L o w e r A i r P r e s s u r e ( A t m )

4. In all-electric homes, the amount of electricity expended is of interest to consumers, builders, and environmental friendly groups involved with energy conservation. Suppose your environmental engineering company is contracted to investigate the monthly electrical usage in all-electric homes and its relationship to the size of the home so that monthly electrical usage can be estimated by the size of a given home. Moreover, suppose you find out that the monthly electrical usage in all-electric homes is related to the size of the home by the quadratic model: y 2 = β0 + β1 x + β2 x + є. To fit the model, your company collects the values for x and y for 10 homes during the month of October. The data is displayed in the table below. Use the method of least-squares to estimate the unknown parameters in the quadratic model. (Note: Make sure to list the initial matrix you used to find these unknown parameters). Furthermore, provide a measure of model fit (coefficient of determination). Size of Home Monthly Usage Size of Home Monthly Usage (ft.2) (kilowatt-hours) (ft.2) (kilowatt-hours) 1,290 1,182 1,350 1,172 1,470 1,264 1,600 1,493 1,710 1,571 1,840 1,711 1,980 1,804 2,230 1,840 2,400 1,956 2,930 1,954

b0 = ______

b1 = ______

b2 = ______

5. A recent article in the Proceedings of the International Conference on Wireless Networks describes a study on a mobile ad hoc computer network. This network consists of several computers (nodes) that move within a network area. Often messages are sent from one node to another. When the receiving node is out of range, the message must be sent to a nearby node, which then forwards it from node to node along a routing path toward its destination. The routing path is determined by a routine known as a routing protocol. The ratio of messages that are successfully delivered to the total messages sent, which is called the goodput, is affected by the average node speed and by the length of time that the nodes pause at each destination. The table shown below presents average node speed, average pause time, and goodput for 25 simulated mobile ad hoc networks. Use the data presented below to fit a full second-order quadratic model with interaction between speed (x1) and pause time (x2) so that the goodput can nd 2 2 be predicted. 2 -order model with interaction: y=β0 + β1 x1 + β2 x2 + β3 x1x2 + β4 x1 + β5 x2 + є. Speed (m/s) Pause Time (s) Goodput Speed (m/s) Pause Time (s) Goodput 5 10 .951 20 40 .878 5 20 .946 20 50 .899 5 30 .947 30 10 .630 5 40 .943 30 20 .761 5 50 .946 30 30 .849 10 10 .908 30 40 .877 10 20 .902 30 50 .906 10 30 .913 40 10 .553 10 40 .913 40 20 .783 10 50 .921 40 30 .846 20 10 .724 40 40 .871 20 20 .821 40 50 .901 20 30 .849 (Note: Make sure to list the initial matrix you used to find these unknown parameters). Ans:

Unknown Parameters: b0 == ______b2 == ______b4 == ______

b1 == ______b3 == ______b5 == ______6. Cardio-respiratory fitness is widely recognized as a major component of overall physical well-being. Direct measurement of maximal oxygen uptake (VO2max) is the single best measure (in L/min) of such fitness, but direct measurement is time consuming and expensive. It is therefore desirable to have a prediction equation for VO2max in terms of easily obtained quantities. A model reported under study in the Research Quarterly for Exercise and Sport for male students is: y = 5.0 + 0.01x1 – 0.05x2 – 0.13x3 – 0.01x4 + є where the four factors being utilized are: x1 = weight (kg), x2 = age (yr), x3 = time to walk 1-mile (min), and x4 = heart beat at the end of the walk (beats/min). What is the probability that VO2max will be between 1.0 and 2.6 for a single observation made when the weight is 76 kg, age is 20, walk time is 12 min, and heart rate is 140 b/m. An estimate of model error standard deviation was calculated at σ = 0.40. Ans:

7. A study reports the accompanying data on discharge amount (q, in m3/sec), flow area (a, in m2), and slope of the water surface (b, in m/m) obtained at a number of floodplain stations. The study proposed a multiplicative power model: Q = α aβ bγ є. Q 15 25 6 3.0 8 A 9 30 6 1.5 2 B .005 .007 .004 .04 .04

Q 88 55 27 10 11 A 41 26 12 7 10 B .007 .006 .004 .003 .003 Use an appropriate transformation to make the model linear and then estimate the regression parameters for the transformed model. Finally, estimate α, β, and γ, (the parameters of the original model). (Be sure to state your initial matrix). Ans:

α = ______β = ______γ = ______THE END