# Sec.14.7 - Designing a dumpster

Applied Project in Sec.14.7, Calculus by Stewart

Chinese version 設計垃圾桶

For this project we locate a rectangular trash Dumpster in order to study and describe all its shape and construction. We then attempt to determine the dimensions of a container of similar design that minimize construction cost.

**Question 1 :**

First locate a trash dumpster in your area. Carefully study and describe all details of its construction, and determine its volume. Include a sketch of the container.

**Answer：**

- Width $x = 0.2 (m)$
- Length $y = 0.25 (m)$
- Height $z = 0.3 (m)$

$$ volume = 0.2 \times 0.25 \times 0.3 = 0.015 (m^3) $$

**Question 2：**

While maintaining the general shape and method of construction, determine the dimensions such a container of the same volume should have in order to minimize the cost of construction. Use the following assumptions in your analysis:

- The sides, back, and front are to be made from 12-gauge ($2.657 (mm)$ thick) steel sheets, which cost $8.00 per square foot (including any required cuts or bends).
- The base is to be made from a 10-gauge ($3.416 (mm)$ thick) steel sheet, which costs $10.00 per square foot.
- Lids cost approximately $50.00 each, regardless of dimensions.
- Welding costs approximately $0.60 per square meter for material and labor combined.

Give justification of any further assumptions or simplifications made of the details of construction.

**Answer：**

From these figures the cost equation of a rectangular dumpster was determined to be:
$$
8\times 2\times (x+y)\times z+10xy+50+0.6\times (2x+2y+4z)
$$
Where the dimensions are based on a 3-dimentional graph, Width=x, Length=y, and Height=z. It is also known that the dumpster volume will have to equal to $0.015 (m^3)$, i.e.,
$$
xyz=0.015\ or\ z=\frac{0.015}{xy}
$$
so the expression for Total Cost $(f)$ becomes
$$
f=0.24\left(\frac{1}{x}+\frac{1}{y}\right)+10xy+50+1.2x+1.2y+\frac{0.036}{xy}
$$
To have a minimum we should have $f_x=f_y=0$ where
$$
f_x=\frac{-0.24}{x^2}+10y+1.2-\frac{0.036}{x^2y}\

f_y=\frac{-0.24}{y^2}+10x+1.2-\frac{0.036}{xy^2}
$$
Since $x$ and $y$ must be positive in this problem, we have $x=y$. We then get $10x^4+ 1.2x^3 -0.24x -0.036=0$ that has only one positive root $x\approx 0.295$.

So we find Total cost is about $53.6.

**Question 3:**

Describe how any of your assumptions or simplifications may affect the actual result.

**Answer:**

Since the length of the side is much bigger then it’s thickness we simply ignore the thickness.

**Question 4：**

If you were hired as a consultant on this investigation, what would your conclusion be? Would you recommend altering the design of the dumpster? If so, describe the savings that would result.

**Answer**

The cost of the original trash dumpster is $53.92, which is larger than the dumpster we design. So we would recommend altering the design of the dumpster, and we can save about $0.3.