Problems involving equal surface area and volume

**200001** · April 5th 2009, 06:23 AM

I have had a few of these questions (both volume and surface area) and not comfortable with changing the formula over. Any help is appreciated to logical steps for this and other such problems

Question1

A cone has radius

r and slant height l.
A cylinder has radius r and height r.

The total surface area of the cone is equal to the total surface area of the cylinder.

Find an expression for l in terms of r.

**earboth** · April 5th 2009, 07:03 AM

Originally Posted by 200001

I have had a few of these questions (both volume and surface area) and not comfortable with changing the formula over. Any help is appreciated to logical steps for this and other such problems

Question1

A cone has radius

r and slant height l.
A cylinder has radius r and height r.

The total surface area of the cone is equal to the total surface area of the cylinder.

Find an expression for l in terms of r.

1. The total surface area of a cylinder is:

$a_{cyl} = 2 \pi r^2 + 2\pi r h$

With h = r you get $a_{cyl} = 4\pi r^2$

2. The total surface area of a cone is:

$a_{cone} = \pi r^2 + \pi r l$

3. Both areas are equal:

$\pi r^2 + \pi r l = 4 \pi r^2~\implies~\boxed{l = 3r}$

**200001** · April 5th 2009, 08:02 AM

Thanks for your help

I am still struggling to make the connection between this part:

With h = r you get

and this part

I am trying to work out how you got to those findings

Regards

**stapel** · April 5th 2009, 10:14 AM

If you plug "r" in for "h" (because h = r) in $a_{cyl}\, =\, 2\pi r^2\, +\, 2\pi rh$ , what do you get? (Plug "r" in for "h" in the formula you were given, simplify, and note the result.)

If you solve $\pi r^2 l\, +\, \pi rl\, =\, 4\pi r^2$ for $l\, =$ , what do you get? (Divide through by $\pi r$ and then solve the resulting literal equation.)

If you get stuck, please reply showing how far you have gotten. Thank you!

**200001** · April 6th 2009, 01:27 AM

Great, Thanks
I understand now
Regards

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