Optimisation: Curved surface area of cone

[Monkeyseat]

Sorry guys, this is the last one, I'm just stumped again and after having a think I can't see where I'm going wrong.

Here is the question:

Here is my working:

I baisically equated the volume equation to the total volume, rearranged for h and input it into the surface area equation. One thing I wasn't sure of though was whether I had to square it to get S^2 like the equation in the book states in the above image. It could need to be squared, but if so, why? I was wondering about this because in a later part of the question I had to get the minimum surface area so I had to square root the answer that the curved surface equation gave me after putting the minimum value of 1.41 in. I didn't realise at first that I had to do that, I thought S^2 represented the area, not S.

Once again however, I can't seem to get the same as the book, where am I going wrong?

Thanks. Sorry for posting it as an image but I find it easier to write it out than type it.

 

[Deleted member 4993]

You need to do the substitution into \(\displaystyle S^2\) instead of S.

 

[Deleted member 4993]

\(\displaystyle S^2 = \pi^2 r^2\cdot (r^2 + h^2)\)

\(\displaystyle S^2 = \pi^2 r^2\cdot (r^2 + \frac{16}{r^4})\)

\(\displaystyle S^2 = \pi^2 \cdot (r^4 + \frac{16}{r^2})\)

Thats it.....

 

[Monkeyseat]

You need to do the substitution into \(\displaystyle S^2\) instead of S.

Thanks for the reply. I saw that just before you replied. Is it an acceptable method if I just square all the terms in my current answer:

Or should I attack it differently from the start and solve for S^2 straight away instead of S first? I saw your other reply, would it be better to start/do it like that? Just wondering.

The pic is a bit big, apologies.

Is S^2 the surface area or is S the surface area? In a later question it asks you to find the minimum surface area and I had to square root the value that the S^2 equation gave me to get the right answer. So is S the area? I'm just not sure what S^2 is for and why it is used by the book when it could be left at S. It says S cm^2 is the volume. Again I saw both your responses and I am grateful but if you know, please could you just clear this up for me?

\(\displaystyle S^2 = \pi^2 r^2\cdot (r^2 + \frac{16}{r^4})\)

\(\displaystyle S^2 = \pi^2 \cdot (r^4 + \frac{16}{r^2})\)

Thats it.....

How did you get to 16/r^2 from 16/r^4? I know that's the answer but I don't know how you changed the denominator there doing what you did.

Any help appreciated.

Thanks.