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* width\text{ }* height`

Substitute the values:

`375=3w* w* h`

Solve for h:

`\begin{gathered} 375=3w^2h \\ (375)/(3w^2)=(3w^2h)/(3w^2) \\ h=(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((125)/(w^2)) \\ S=3w^2+(1000)/(w) \end{gathered}`

Answer:

`S(w)=3w^2+(1000)/(w)`