Question

I’m having trouble understanding how to solve this problem….A company wants to construct an open rectangular box with a volume of 375in^3 so that the length of its base is 3 times the with. Express the surface area, S, of the box as a function of the with w. Also find the domain.The answer I got is in the picture

Answer 1

We have:

Volume = V = 375in^3

Length = 3w

Width = w

Area = S

Height = h

Then, the formula of the volume is given by:

$V=length\times width\text{ }\times height$

Substitute the values:

$375=3w\times w\times h$

Solve for h:

$\begin{gathered} 375=3w^2h \\ \frac{375}{3w^2}=\frac{3w^2h}{3w^2} \\ h=\frac{125}{w^2} \end{gathered}$

Next, the surface area, S, of the box is:

$\begin{gathered} S=area\text{ of base}+2area\text{ vertical side + 2area other vertical side} \\ S=3w(w)+2(3w)(h)+2(w)(h) \end{gathered}$

Simplify:

$S=3w^2+6wh+2wh=3w^2+8wh$

Substitute the value of h:

$\begin{gathered} S=3w^2+8w(\frac{125}{w^2}) \\ S=3w^2+\frac{1000}{w} \end{gathered}$

Answer:

$S(w)=3w^2+\frac{1000}{w}$