Question

problem on mathematical modelling in engineering (please answer 4(a) i ,ii and (b))

4. Lumber Cutters-Lumber cutters wish to use readily available measurements to estimate the number of board feet of lumber in a tree. Assume they measure the diameter of the tree in inches at waist height. Develop a model that predicts board feet as a function of diameter in inches. Use the following data for your test: 17 19 20 23 25 28 32 38 39 41 X y 19 25 32 57 71 113 123 252 259 294 The variable x is the diameter of a ponderosa pine in inches, and y is the number of board feet divided by 10. a. Consider two separate assumptions, allowing each to lead to a model. Completely analyze each model. i. Assume that all trees are right-circular cylinders and are approximately the same height. ii. Assume that all trees are right-circular cylinders and that the height of the tree is proportional to the diameter. b. Which model appears to be better? Why? Justify your conclusions. I

Answer 1

Given that Lumbess cutters Lumbes Catters wish to use available measwements to estimate the endily boaxd feat of lumbes in a tse mumbes measre the diamdex of the trea in incha AsSame modd tha predics hght Deldep at (naist function of diamades o ncha boasds fet as a 17 X 19 20 93 25 28 32 38 39 41 Y19 259 299 123 25 32 57 I13 252 Pardeso sa pine boasd feot is the iamdey of The Vasrable numbes the in inches, by divided l0 as the volumo (a) The numbes of boond fat Under th tirt assumption t)that all tras ase and are igt- Cirula ylndas we have thaltho Noluue of Sano the tree is the No Ah

A is-h avera Cess Sadion ana of the tre &histhe height . Sine e ase asSuming tha al taes ane-he Same haight, VocA The Conss Sedina Qsea is e is tesos of the aius Sauaed c0). Ld d is th chaaclesstie climensien Th) We hale A ocatel d So finaly elating he ol to the chanderctic dimersion Ved2 Now,te ddesmjins the Popotyomali ty Cnstad &it this assumphion Sems easonble

1ю до- гкю анм - L00 1ю- о- о го «о соо воо 1000120 LO1E00 10 з0 -Я

n -1-5191e673Gd to -the sae dh

(3) The Second assumption (i), tha all tres ave cqlinders and tha the Propational -the tree is to the diamedes, we have that the volume ofthe tvee is V A A is ave Sediona the tre Asea C 5 as aboe. undes this SeCond to he dams h is the assimpt on, haigh held is prepotiona A ad Vod(d2)- d3 to detesmine the Prpotienaly constantg No is this assumption Sems oea sonale Ne the given data

30 2000- 200 1M 1,00 o1a20,00034000s0 6a00700 10 20

oSinq Least s d o.04362034 724x = 0.04362034124d to the auw dak.