Question

A cardboard box has a square base with each edge of the box having length of x inches. The total length of all 12 edges of the box is 144in. Express the volume of the box as a function of x. Express the height in terms of x.

Answer 1

Let l, b and h be the length , breadth and height of cuboidal box.
Since, the box has a square base, l = b = x inches

$$\begin{gathered} \mathrm{length}\mathrm{of}12\mathrm{edges}=144\mathrm{inches} \\ \Rightarrow 4(l+b+h)=144 \\ \Rightarrow (l+b+h)=36 \\ \Rightarrow x+x+h=36 \\ \Rightarrow 2x+h=36 \\ \Rightarrow h=(36-2x)\mathrm{inches} \end{gathered}$$

$$\begin{gathered} \mathrm{volume}\mathrm{of}\mathrm{box}=l\times b\times h \\ =x\times x\times (36-2x) \\ =x^2(36-2x)=\left(36x^2-2x^3\right)\mathrm{cubic}\mathrm{inches} \end{gathered}$$