Theproject will be carried out by first identifying a dumpster in theresidential area and all its dimensions measured and recorded. Themain aim of the project is to design a cost-efficient dumpster withthe same volume. The design will be carried out basing on Chapter14.7 of the textbook on establishing minimum and maximum values. Theproject entails developing a total cost equation, calculating partialderivatives and determining the critical points.
Calculatingthe volume of the dumpster
Volumeof the dumpster= Total area of faces * height of dumpster
Findingthe area of the faces:
(referto the diagram of dumpster at the end of this project for dimensions)
Areaof the first face (i) = 0.5*9*9 = 40.5
Areaof the second face (ii) = 9*63 = 567
Areaof the third face (iii) = 72*43.5 = 3132
Areaof the fourth face(IV) = 14.5*29.5 = 427.75
Areaof the fifth face (v) = .5*42.5*34.9 = 71.6
Areaof the sixth face (VI) = 7.7*29.5*.5 = 113.58
Totalarea = 5022.43
Volumeof the dumpster (V):
V=5022.43*72 = 361,615 in3or209.27 ft­3
Determiningthe total cost:
Thesides, back and front are created from 12-gauge steel material whichcosts $0.70 per square foot, the base will be constructed from10-gauge steel material estimated to costs $0.90 per square foot alid cost $50 and the welding cost per foot for material inclusiveof labor will be $0.18.
T=1.4xz + 1.4yz + .9xy + .32x + .32y + .64z + 50
x=Width,y=Length, and z=Height. (All the dimensions are based on a3-dimentional graph)
Thedumpster volume= 209ft3,Hence:
xyz= 209 or z  =  209/(xy)
Reducingthe equation into 2 variables, z value will be substituted into thetotal cost equation.  Thus:
OverallCost = 1.4(209/y) QUOTE +1.4(209/x)+0.9xy+0.32x+0.32y+0.64(209/(xy))+50
Determiningpartial derivatives of the total cost equation
Fromtotal cost equation, we get partial derivative fxand fy.
Determiningthe critical points
First,partial derivative equations are simplified and equated to zero.
Then,both sides are multiplied by 10 and subtracting gives:
(0=-2926y+9x2y2+3.2x2y-1337.6) – 0=-2926x+9 x2y2+3.2y2x-1337.6)
0=-2926y+2926x+3.2 x2y-3.2 y2x
Furthersimplifying the resulting equation and equating it to 0, we finallyget:
(2926+3.2xy)(x-y)=0,2926+3.2xy=0 0r x-y= 0 where (x=y)
Thesolution, x=-2926/ (3.2y) is not feasible. For a solution to befeasible x and y must be greater than zero.
Thevalue of z= 6ft is selected to get a feasible solution. Hence, aftersubstituting in z=209/ (xy), the new values of x=y, will be 5.901.Using the above values the estimated total cost will be $ 188.12.
Acrucial factor to consider in this study is that a dumpster of arectangular shape was used to solve a minimizing equation. Theapproach utilized in this particular design is much more expensive ascompared to the original plan and a Construction firm may find itunfeasible to use this model.
Diagrammaticrepresentation of a dumpster