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,

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

Derivate C, and find its minimum, as shown below

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

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