C2 - Differentitation - Solving Practical Problems HELP

Hi,

I'm stuck on this question, which is on solving practical problems using differentitation.

The slant height of the cone shown is 18cm. (The diagram doesn't give any other information)

a) Find the volume of the cone in terms of h. I did this and got V=1/3'pi'r²h

b) Hence find the exact value of the maximum volume of the cone.

I wrote h in terms of r, by using pythagoras, 18²=h²+r² and got √324-r²

So the equation I've ended up with is V=1/3'pi'r²√324-r² and I really don't know what to do next.

The answer to part b) is 423'pi'√3cm²

Thank you very much.

Reply 1

jalapeno

*pi

Reply 2

Dekota

sweet_gurl

The slant height of the cone shown is 18cm.

a) Find the volume of the cone in terms of h. I did this and got V=1/3'pie'r²h
b) Hence find the exact value of the maximum volume of the cone.
The answer to part b) is 423'pie'√3cm²

π = pi

a) V=1/3πr²h = (πr²h)/3

If you draw a picture of the cone, the slant height is 18cm, so notice the relationship:

18² = r² + h²
re-arrange: r² = 18² - h²

Now substitute r into the Volume equation >>> V = (π(18² - h²))/3
................................................................V = (18²π - πh²)/3

Now differentiate to get dV/dh = -(h²-108)π

Now sub dV/dh = 0 >>>> -(h²-108)π = 0

h = √108

sub h=√108 into (18² = r² + h²) >>>> 18² = r² + 108
r = √216

sub h=√108 and r = √216 into original Volume equation ((πr²h)/3) >> (π(√216)²√108)/3
= (216π√108)/3cm³

This answer is close to the answer that you state, but not exact.
Plus I notice you state the volume as cm², this cannot be a volume, but an area.