Designinga Dumpster

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Designinga Dumpster

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)

Area=length *width

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 in^{3}or209.27 ft­^{3}

*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.

*TotalCost Equation:*

T=1.4xz + 1.4yz + .9xy + .32x + .32y + .64z + 50

Where:

T=Total cost

x=Width,y=Length, and z=Height. (All the dimensions are based on a3-dimentional graph)

Thedumpster volume= 209ft^{3},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 f_{x}and f_{y}.

f_{x}=-292.6/(x^{2})+0.9y+0.32-133.76/(yx^{2})

f_{y}==-292.6/(y^{2})+0.9y+0.32-133.76/(xy^{2})

*Determiningthe critical points *

First,partial derivative equations are simplified and equated to zero.

f_{x}, 0=-292.6y+0.9x^{2}y^{2}+0.32x^{2}y-133.76

f_{y}.0=292.6x+0.9x^{2}y^{2}+0.32y^{2}x-133.76

Then,both sides are multiplied by 10 and subtracting gives:

(0=-2926y+9x^{2}y^{2}+3.2x^{2}y-1337.6) – 0=-2926x+9 x^{2}y^{2}+3.2y^{2}x-1337.6)

0=-2926y+2926x+3.2 x^{2}y-3.2 y^{2}x

Furthersimplifying the resulting equation and equating it to 0, we finallyget:

2926(x-y)+3.2xy(x-y) =0

(2926+3.2xy)(x-y)=0,2926+3.2xy=0 0r x-y= 0 where (x=y)

*Testingfor feasibility*

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*