Q. 53 - Precalculus Enhanced with Graphing Utilities

Question

Minimizing Surface Area United Parcel Service has

contracted you to design a closed box with a square

base that has a volume of 10,000 cubic inches. See the

illustration.

(a) Express the surface area S of the box as a function of x.

(b) Using a graphing utility, graph the function found in

part (a).

(c) What is the minimum amount of cardboard that can be

used to construct the box?

(d) What are the dimensions of the box that minimize the

surface area?

(e) Why might UPS be interested in designing a box that

minimizes the surface area?

Step-by-Step Solution

Answer

(a) The surface area of the box as a function of x is:

$\frac{2x^3+40,000}{x}$

(b) 

(c) $2894.43 Rs.$

(d) $l=b=x=10\sqrt{20}, h=y=5$

(e) Because, when the surface will be minimum it will minimize the cost of designing the box.

Step 1. Given information

given,

Minimizing Surface Area United Parcel Service has

contracted you to design a closed box with a square

the base that has a volume of 10,000 cubic inches.

according to the question,

length=l=x, breadth=b=x, height=h=y

Step 2. Express the surface area S of the box as a function of x.

The surface area of cubiod= $2(lb+bh+hl)$

The surface area S of the box =$2(x^2+2xy)=2x^2+4xy$   .....(1)

Volume of cubiod=$lbh$=$x^{2}y$

Volume of box=$$\begin{gathered} 10,000=x^{2}y \\ y=\frac{10,000}{x^2} \end{gathered}$$

put the value of y in equation 1 then,

S= $2x^2+4x\frac{10,000}{x^2}=2x^3+\frac{40,000}{x}$

Thus, the surface area S of the box as a function of x=$\frac{2x^3+40,000}{x}$

Step 3. The minimum amount of cardboard that can be used to construct the box

total Area =$x^2+4xy$               ....(2)

Volume of box=$10,000=x^{2}y$

$y=\frac{10,000}{x^2}$

put the value of y in $e{q}^n 2$ 

Area=$x^2+\frac{40,000}{x^2}$                  ....(3)

On solving $$\begin{gathered} e{q}^n 3 we find value of x, \\ x=10\sqrt{20} \end{gathered}$$

and $y=\frac{10,000}{{(10\sqrt{20} )}^2}=\frac{100}{20}=5$

Minimum cost=$$\begin{gathered} x^2+4xy={(10\sqrt{20)}}^2 +4\times 10\sqrt{20}\times 5 \\ =2894.43 \end{gathered}$$

Step 4. The dimensions of the box that minimize the surface area

$$\begin{gathered} l=b=x=10\sqrt{20} \\ h=y=5 \end{gathered}$$

Step 5 Why might UPS be interested in designing a box that minimizes the surface area?

Because, when the surface will be minimum it will minimize the cost of designing the box.

So UPS might be interested in designing a box that minimizes the surface area.