Question

Please help me solve any part of this im just at a loss

Answer 1

The volume of the box, since it has a square base, is given by the formula below

`\begin{gathered} V=l^2*h \\ l\rightarrow\text{ side of the square base} \\ h\rightarrow\text{ height of the box} \end{gathered}`

Then, according to the question,

`\Rightarrow l^2h=1200`

Additionally, the cost of the box depends on its total surface area; then,

`C=2l^2+4lh`

Therefore, using the first equation on C,

`\begin{gathered} \Rightarrow h=(1200)/(l^2) \\ \Rightarrow C=2l^2+4l((1200)/(l^2))=2l^2+(4800)/(l) \end{gathered}`

Derivate C, and find its minimum, as shown below

`\begin{gathered} C^(\prime)(l)=4l-(4800)/(l^2) \\ \Rightarrow C^(\prime)(l)=0 \\ \Rightarrow4l-(4800)/(l^2)=0 \\ \Rightarrow4l=(4800)/(l^2) \\ \Rightarrow4l^3=4800 \\ \Rightarrow l^3=1200 \\ \Rightarrow l=10.62 \end{gathered}`

The answer is 10.6inches. (Side of the square base).