Question

Write an equation for S as a function of w when the volume of the prism is 125 in^3. Explain or show your work.

Answer 1

The volume of a prism is given by the formula:

`V=w* l* h`

Where w is the width, l is the length and h is the height of the prism. In this case, we are dealing with a square prism, which means that the base of the prism is a square, as you know, the length and width of a square are the same, thus l=w, then we get:

`V=w* l* h=w* w* h=w^2* h`

To find the area of the prism, we have to find the area of each face and then sum them up.

the faces are either a rectangle or a square, for this kind of figures the area equals the length times the width of the figure:

Then, for the top and bottom faces, the area is:

`A1=w* w=w^2`

For the right and left faces, we have:

`A2=w* h`

For the front and back faces we have:

`A3=w* h`

By summing up these areas, we get:

`\begin{gathered} S=A1+A1+A2+A2+A3+A3 \\ S=w^2+w^2+wh+wh+wh+wh \\ S=2w^2+4wh \end{gathered}`

Where we added 2 times each area because opposite areas have the same area (like the bottom and top faces, both areas are equal)

Now that we have two expressions, for the area and the volume of the prism, we need to write an equation of S as a function of w, this means that we have to find an expression to calculate S with only one variable appearing (w) so far in our formula of S we have the variables w and h, then we need to find the way of writing h as a function of w and then replace it into the formula of S.

If we take the formula of the volume, replace 125 in^3 for V and then solve for h, we get:

`\begin{gathered} V=w^2h \\ 125=w^2h \\ (125)/(w^2)=(w^2)/(w^2)h \\ (125)/(w^2)=1* h \\ h=(125)/(w^2) \end{gathered}`

Now let's substitute this expression into the formula of S, like this:

`\begin{gathered} S=2w^2+4wh \\ S=2w^2+4w*(125)/(w^2) \\ S=2w^2+4*125*(w)/(w^2) \\ S=2w^2+500(1)/(w) \end{gathered}`

Then, the equation of S as a function of w is:

`S=2w^2+(500)/(w)`