Word Problem: Find radius, height of box w/ max volume

[Carla714]

The problem is: A cylindrical box will be tied up with ribbon as shown in the figure (figure shows a box with ribbon crossing it twice - ribbon crossing in an "x" in the middle and a bow). The longest piece of ribbon is 130 cm long and 10 cm of that are required for the bow. Find the radius and height of the box with the largest possible volume.

I know that the formula for volume is V=pi*r^2h. I figure that 4*2r + 4h =120. But I am stuck from there.

Help please?

 

[galactus]

You're in good shape with your formula so far.

Now, \(\displaystyle 8r+4h=120\)

We can solve for r or h. Let's solve for h and get:

\(\displaystyle h=30-2r\)

Sub into the volume formula:

\(\displaystyle \L\\{\pi}r^{2}(30-2r)=30{\pi}r^{2}-2{\pi}r^{3}\)

Now, differentiate, set to 0 and solve for r. h will follow. Hint: h and r will be the same.

 

[Carla714]

Thanks so much! I had solved for h and started putting it in the volume formula but wasn't confident I was going in the right direction! Thanks for the boost!