related rates question (1 Viewer)

[stag_j]

A vessel is in the shape of a right circular cone with vertex downwards and axis vertical; the height of the vessel is 12m and the radius of its circular top is 8m. The vessel is being filled with water, the inflow of water is at the uniform rate 2m^3/min.
When the height if water is 'h' meters, find an expression for the radius 'r' meters of the water surface in terms if 'h', and prove that the volume V m^3 of water is V = (4pih^3)/27.
At what rate is the surface of the water rising when its depth is 3m?

Generally i can do related rates questions fine, but i'm having trouble starting this. Given that it is a right circular cone, I would have thought the the height should be the same as the radius...but the question states that h=12 and r=8, so i struggle to see how it is still a right circular cone...

 

[wogboy]

A right circular cone doesn't necessarily have to have it's radius equal to its height. All it means is that the line joining the vertex of the cone and the centre of the base of the cone, is perpendicular to the base of the cone (i.e. if you put the cone down on it's base, the vertex of the cone is directly above the centre of the base).

In this cone, you can see that if the water level has height h, this means that the radius r of the "water cone" will be linearly proportional to h. Also you know that when h=0, r=0 and also when h=12m, r=8m.

then h = 3*r/2
and r = 2*h/3

also the volume V of the cone formed by the water, will be:

V = (1/3)*pi*(r^2)*h
= (1/3)*pi*((2/3)^2)*h^3
= (4/27)*pi*h^3

hence dV/dh = (4/9)*pi*(h^2)

you're given that dV/dt = 2 m^3/min (inflow rate)
and also you know that:
dh/dt = dh/dV * dV/dt
so
dh/dt = 9/(4*pi*h^2) * 2
= 9/(2*pi*h^2)

so when h = 3m, dh/dt = 9/(2*pi*9) = 0.16 m/min = 0.27 cm/s

 

[stag_j]

yes i realised that i'd misinterpreted the question wrong. i had been thinking that the question implied that the angle at the vertex was a right angle. i asked my teacher about it and he said that it was a bad question. but when i got home i thought about it and realised we had both been wrong.
kinda scares me a little tho - head teacher of maths should have picked up on that i would have thought...