Q. 40 - Precalculus Enhanced with Graphing Utilities

Question

Constructing a Closed Box A closed box with a square base is required to have a volume of $10$ cubic feet.

(a) Build a model that expresses the amount $A$ of material used to make such a box as a function of the length $x$ of a side of the square base.

(b) How much material is required for a base $1$ foot by $1$ foot?

(c) How much material is required for a base $2$ feet by $2$ feet?

(d) Graph $A=A\left(x\right)$. For what value of $x$ is $A$ smallest?

Step-by-Step Solution

Answer

(a) The amount $A$ of material used to make such a box is $A\left(x\right)=2x^2+\frac{40}{x}$

(b) Material required for a base $1$ foot by

$1$ foot is $42$ foot².

(c) Material required for a base $2$ foot by

$2$ foot is $28$ foot².

(d) The value of $A$ is smallest at $\sqrt[3]{10}$ feet.

Step 1. Part (a) Build a model that expresses the amount A of material used to make such a box as a function of the length x of a side of the square base.

Let $h$ be the height of the box and $x$ be the side of a square base.

Volume$=$base area$\times$height

$$\begin{gathered} V=x^{2}h \\ h=\frac{V}{x^2} \end{gathered}$$

Since $V=10$, we get

$h=\frac{10}{x^2}$

Now the amount $A$ of material used to make such a box is

$$\begin{gathered} A=2\times area of the base+4\times area of the side \\ A=2x^2+4hx \end{gathered}$$

Since $h=\frac{10}{x^2}$,

$$\begin{gathered} A\left(x\right)=2x^2+4\left(\frac{10}{x^2}\right)x \\ A\left(x\right)=2x^2+\frac{40}{x} \end{gathered}$$

Step 2. Part (b) Find material required for the box of base 1 foot by 1 foot.

$$\begin{gathered} A\left(1\right)=2{\left(1\right)}^2+\frac{40}{1} \\ A\left(1\right)=2+40 \\ A\left(1\right)=42 foo{t}^2 \end{gathered}$$

Step 3. Part (c) Find material required for the box of base 2 foot by 2 foot.

$$\begin{gathered} A\left(2\right)=2{\left(2\right)}^2+\frac{40}{2} \\ A\left(2\right)=8+20 \\ A\left(2\right)=28 foo{t}^2 \end{gathered}$$

Step 4. Part (d) Draw the graph of A ( x ) and find the value of x where A is smallest.

The graph is

The value of $x$ where $A$ is smallest is $x=2.15$ feet(accurate value is $\sqrt[3]{10}$feet)