D. Keffer, MSE 30, Dept. of Materials Science & Engeerg, University of ennessee, Knoxville Homework ssignment Number hree Solutions Problem. chemical plant produces thousands of s of -plus Liquid Fungicide and thousand of s of -Gone Liquid Insecticide per month. he two processes share some raw materials and facilities so that the amount of and produced are not dependent of each other. In fact the amount of produced is related to the amount of produced by 40 where is the amount of raw materials received at the plant a given month (also s). he total amount of product thousands of s is given as (, ) he monthly production schedule for 0 is as followed Month (thousand of s) (thousands of s) Jan 50 0 Feb 50 0 Mar 60 0 pr 70 0 May 80 30 Jun 90 30 Jul 00 30 ug 00 30 Sep 90 30 Oct 80 0 Nov 70 0 Dec 60 00 In all problems: PU UNIS WIH NSWERS OR YOU WILL NO REEIE FULL REDI. In all relevant problems: WRIE DOWN HE FOUL YOU USE, EFORE YOU USE I. (a) Is this problem contuous or discrete? he problem is discrete. here are elements each sample space. What are the PDFs for the variables and? he PDF s for and are F ( ) and F ( ) (b) Fd the average monthly production of. E ( ) xf ( x) x
D. Keffer, MSE 30, Dept. of Materials Science & Engeerg, University of ennessee, Knoxville E( ) 75s (c) Fd the average monthly production of. is just a function of and. h( x, y) (, ) E ( h(, Y )) h( x, y) f ( x, y) x y E ( (, )) (, ) f ( ) 4 (, ), (d) Fd the average monthly usage of. E ( ) xf ( x) x 3.3s E( ) 0.8s (e) Fd the mean of the total monthly production,. E( a b) ae( ) b so E ( ) E( ) E( ) E( ) 75 3.3333 98.3 s (f.) Fd the variance of the monthly production of usg the rigorous defition of the variance. E f ( ) Month () f f ( ) Jan 50 65 0.0833 5.08333 Feb 50 65 0.0833 5.08333 Mar 60 5 0.0833 8.75 pr 70 5 0.0833.083333 May 80 5 0.0833.083333 Jun 90 5 0.0833 8.75 Jul 00 65 0.0833 5.08333 ug 00 65 0.0833 5.08333 Sep 90 5 0.0833 8.75 Oct 80 5 0.0833.083333 Nov 70 5 0.0833.083333 Dec 60 5 0.0833 8.75 = 75 = 9.67
D. Keffer, MSE 30, Dept. of Materials Science & Engeerg, University of ennessee, Knoxville 9.7 s squared (f.) Fd the variance of the monthly production of usg the mean of the squares mus the square of the mean formula. E E E E 596.7 75 9. 7 his is the same result as was obtaed part (f.). s squared (g) Fd the variance of the monthly usage of. E 469.7 0.8333 9. 0 E s squared (h.) Fd the variance of the monthly production of from tabulated values of. Once the values of, have been tabulated. You can consider it as a random variable itself, rather than as a function of and. E 56.5 3.3333 5. 4 E s squared (h.) Fd the variance of the monthly production of from the variances of and and the formula for given this problem statement. o determe the variance of the function (, ), 40 we need to recall the rules of the variance of a lear combation of Rs: a and a by a b Y ab a b Now, the first statement tells us that the variance of the constant 40 is zero. Furthermore, we can see how to handle the variance of the -/ front of. he second equation tells us that we need to calculate, addition to the variance of and, the covariance of,. he covariance is given by E E EY Y we know the average of and the average of, but we need the average of the product of and. E( ) 975 s squared Now that we have this mean, we can calculate the covariance: 3
D. Keffer, MSE 30, Dept. of Materials Science & Engeerg, University of ennessee, Knoxville E EE 975 75(0.8333).5s squared. Now, that we have the covariance, we can obta the variance of the function : / 40 90.97 9.6667.5 5.4 s squared. his is the same result as was obtaed part (h.). (i) Fd the variance of the total monthly production,. o determe the variance of the function (, ), (, ) we need to recall the rules of the variance of a lear combation of Rs: a a b ab a b and a by Y So he second equation tells us that we need to calculate, addition to the variance of and, the covariance of,. he covariance of and is given by E EE we know the average of and the average of, but we need the average of the product of and. E( ) 96.67 s squared Now that we have this mean, we can calculate the covariance: E EE 96.67 75(3.3333) 33.33s squared. 4
D. Keffer, MSE 30, Dept. of Materials Science & Engeerg, University of ennessee, Knoxville Now, that we have the covariance, we can obta the variance of the function : 9.6667 5.4 * 33.33 76.4 s squared. (j,k,l,m) Fd the standard deviations of,, and. 9.7 7. s 5.4 7. s 9.0 9.5 s 76.4 6.63 s (n) Fd the covariance of and. In part (i), we determed the covariance to be: E EE 96.67 75(3.3333) 33.33 s squared. (o) Fd the correlation coefficient of and Y so 33.33 9.7 5.4 0.7 (p) Give a physical description of what the value and sign of the correlation coefficient means. () he negative sign of the correlation coefficient dicates that as creases, decreases. his makes sense sce and are competg for the same raw materials. (We can also see this from the defition of the problem statement.) () he value of (remember ) dicates that and are relatively strongly correlated. We know from the defition of, that fact and are learly related. 5
D. Keffer, MSE 30, Dept. of Materials Science & Engeerg, University of ennessee, Knoxville (extra) he fact that is not - means that the monthly fluctuation of the raw material,, is large enough that disrupts much evidence of the actual lear relationship between and. If the Raw Material supply,, was a constant, then. Problem. chemical plant contas a jacketed vessel which the followg isomerization reaction takes place: he rate of the production of, r [/hour], is given by r k is the concentration of [/] and the reaction rate constant, k [s/hour], is given as a where function of the temperature, [Kelv], as k 0.0e 0,000 R where R is the gas constant [8.34 J/mole/K]. his (highly ideal) jacketed vessel keeps temperature perfectly constant at the set temperature of 400 K. he concentration the tank is obtaed from the mass balance accumulation out generation d dt Q, Q k where Q is the volumetric flowrate [s/hour], and has a numerical value of Q 9. 0 l/hour. is the reactor volume, 00. 0 s. Rearrangement yields: Q, Q k d dt and where, is the let concentration of, 0 mole/. We can tegrate this equation to yield k, ln ( Q k) Q, Q k We can rearrange this equation to give us Plot, ( t) Q ke ( Q k) Qk t and on one graph and plot,. t,. r as functions of t for hour 0 t 4. Remember, 6
D. Keffer, MSE 30, Dept. of Materials Science & Engeerg, University of ennessee, Knoxville and (/).0.5.0 0.5 0.0 0.0 6.0.0 8.0 4.0 r (/hour).00.95 r.90.85.80 0.0 6.0.0 8.0 4.0 time (hour) time (hour) We want the average concentration of reactant durg that first day of operation. Our formula for the mean over the range a to b is: b h( x) E ( h( )) h( x) f ( x) dx a For our problem at hand, identify x, a, b, h (x), and f (x). Solution: x is t, a is 0, b is 4, h(x) is (t), f ( x) b a 4 (c) What is the average concentration of reactant, Solution: Qk t, ( t ) Q ke ( Q k) 0 ( t) Qk, ke Qt 4( Q k) ( Q k), durg that first day of operation? 4 dt 0 Qk 4, ke k 4Q 0 4( Q k) ( Q k) ( Q k).3973 6 0e 0.877089 0 (d) What is the average rate of production, r, durg that first day of operation? r k 0,000 0,000 t R 8.34400 k 0.0 e 0.0 e 0.9888 In this problem, k, is a constant. hen usg the rules of lear operators (and the answer from part (c)): Er Ek ke ( 0.9888)(.877089). 8560 hour 4 0 7
D. Keffer, MSE 30, Dept. of Materials Science & Engeerg, University of ennessee, Knoxville (e) What is the average concentration of, In this problem, E E,, durg that first day of operation?,, is a constant. hen usg the rules of lear operators (and the answer from part (c)): E E E,,,.877089 0. 9 E (f) What is the variance of durg that first day of operation? Solution: We know the variance of a function is given by: g Eg g ( x) E We have already calculated the second term on the right hand side part (c). We must calculate the first term on the right hand side. Qk t, E g Q ke dt ( Q k) 0 4 0 E Q k Q k t, t g Q Qke ke 4 ( Q k) 0 E g 4, Q ( ) Q k g 3.55934056 E E Qk t Qke t ( Q k) dt Qk t k e ( Q k) t 4 t 0 E 3.55934056.877089 0.0047 (g) What is the variance of r durg that first day of operation? r k In this problem, k, is a constant. hen usg the rules of lear operators (and the answer from part (f)): E k Ek k E k E k r r k 0.9888 0.0047 0.004 hour (h) What is the variance of durg that first day of operation? Solution:, 8
D. Keffer, MSE 30, Dept. of Materials Science & Engeerg, University of ennessee, Knoxville Remember, the rule for lear operators: a b ab so Now a by Y, ( ),,, is a constant so its variance and covariance are zero. 0 ( ) 0 0.0047 Problem 3. private pilot wishes to sure his airplane for $00,000. he surance company estimates that a total loss may occur with a probability of 0.00, a 50% loss with probability 0.0 and a 5% loss with a probability of 0.. Ignorg all other partial losses, what premium should the surance company charge each year to realize an average profit of $500? solution: he formation given: Insure the plane for $00,000. P(total loss)=0.00 P(half loss)=0.0 P(quarter loss)=0. P(no loss)=remader=-0.-0.0=0.00=0.888. What premium should an surance company charge for an average annual profit of $500? Fd average loss per year. E ( h( x)) h( x) f ( x) Let h(x) be the amount paid. x $ 0 (0.88) $00,000 (0.00) $00,000(0.0) $50,000 (0.) $6400 herefore the annual premium should be $500 dollars more than the mean or: $6900. 9