cubics

[tupac3270]

A cylinder is generated by rotating a rectangle with perimeter 12 inches about one of its sides.
a. write a function V(x) to model the volume of the box using the appropriate lengths.

(In the diagram it shows that the circumference of the base of the cylinder is x)

Please help. I have been working on this problem for 3 hours straight and I am stuck. I do not understand the wording of the question. Thanks

P.S PLEASE HELP ME!!!!!!!!!

 

[Gene]

It is pecular problem.
The rectangle is h high by w wide.
2h+2w=12
The circumfrence x = 2*pi*w
The volume is pi*w^2*h
Solve the first two for h and w in terms of x and substitute into the third.

 

[soroban]

Hello, tupac3270!

A cylinder is generated by rotating a rectangle with perimeter 12 inches about one of its sides.

Write a function V(x) to model the volume of the box using the appropriate lengths.

(In the diagram it shows that the circumference of the base of the cylinder is x)

Are you sure it's the circumference? . . . It makes the problem quite tricky.

A radius of \(\displaystyle x\) makes the problem "nicer".

         x
      *-----*
      |     |
      |     |
  6-2x|     |6-2x
      |     |
      |     |
      *-----*
         x

If this rectangle is rotated about its left side,
\(\displaystyle \;\;\)the radius is \(\displaystyle r\,=\,x\) and the height is \(\displaystyle h\,=\,6\,-\,2x\)

The volume of a cylinder is: \(\displaystyle \,V\:=\:\pi r^2h\)

So we have: \(\displaystyle \,V(x)\;=\;\pi x^2(6\,-\,2x)\)