Question

Find the value of x which optimises the total surface area of the box, and showthat it minimises the total surface area.

Answer 1

Given data:

Let the width of the box be

`=x`

The volume of the box is

The height of the box is

`=h`

Let the length of the box be

`=l`

The length is four times the width of the base, this can be represented below as

`\begin{gathered} l=4* x \\ l=4x\ldots\ldots\ldots\text{.}(1) \end{gathered}`

Part A:

Show that the height h of the box is given by

`(400)/(x^2)`

Concept:

To show that the height is given as above, we will use the volume of the box which is given below as

`\begin{gathered} V_{\text{box}}=\text{length}* width* height \\ V_{\text{box}}=l* x* h \end{gathered}`

By substituting the values, we will have

`\begin{gathered} V_{\text{box}}=l* x* h \\ \text{Substituite equation (1) in the formuka above,} \\ 400=4x* x* h \\ 400=4x^2h \\ \text{divide both sides by 4x}^2 \\ (4x^2h)/(4x^2)=(400)/(4x^2) \\ h=(100)/(x^2)(\text{PROVED)} \end{gathered}`

Part B:

Show that the total surface area A of the box is given by

`A=4x^2+(1000)/(x)`

Concept:

To prove the above relation, we will use the formula of the area of the box below

`\begin{gathered} A_{\text{box}}=2(lw+lh+wh) \\ \text{where,} \\ l=\text{length}=4x \\ w=\text{width}=x \\ h=\text{height}=(100)/(x^2) \end{gathered}`

By substituting the values, we will have

`\begin{gathered} A_{\text{box}}=2(lw+lh+wh) \\ A_{\text{box}}=2(4x* x+4x*(100)/(x^2)+x*(100)/(x^2)) \end{gathered}`

By simplifying the relation above, we will have

`\begin{gathered} A_{\text{box}}=2(4x* x+4x*(100)/(x^2)+x*(100)/(x^2)) \\ A_{\text{box}}=2(4x^2+(400)/(x)+(100)/(x)) \\ A_{\text{box}}=2(4x^2+(500)/(x)) \\ A_{\text{box}}=8x^2+(1000)/(x) \end{gathered}`

Hence,

The Total surface area of the box will be given below as

`A_{\text{box}}=8x^2+(1000)/(x)`

To determine the value of x which optimizes the total surface area of the box, and show

that it minimizes the total surface area, we will have to look for the first derivative of the function above

`\begin{gathered} A_{\text{box}}=8x^2+(1000)/(x) \\ (dA)/(dx)=(d)/(dx)(8x^2+(1000)/(x)) \\ (dA)/(dx)=16x-(1000)/(x^2) \end{gathered}`

To find the value of x, we will substitute the value of dA/dx to be = 0

`\begin{gathered} (dA)/(dx)=0 \\ 16x-(1000)/(x^2)=0 \\ 16x=(1000)/(x^2) \\ (16x^3)/(16)=(1000)/(16) \\ x^3=62.5 \\ x=\sqrt[3]{62.5} \end{gathered}`

To show the minimum value of x, we will have to look for the second derivative

`\frac{d^2A^{}}{dx^2}`