# Designing a Dumpster Affiliated essay

Designinga Dumpster

Affiliated

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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

&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbspVolumeof 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 in3or209.27 ft&shy3

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.&nbsp(All the dimensions are based on a3-dimentional graph)

Thedumpster volume= 209ft3,Hence:

xyz= 209 or z&nbsp =&nbsp 209/(xy)

Reducingthe equation into 2 variables, z value will be substituted into thetotal cost equation.&nbsp 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.

fx=-292.6/(x2)+0.9y+0.32-133.76/(yx2)

fy==-292.6/(y2)+0.9y+0.32-133.76/(xy2)

Determiningthe critical points

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

fx, 0=-292.6y+0.9x2y2+0.32x2y-133.76

fy.0=292.6x+0.9x2y2+0.32y2x-133.76

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(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