Rates of Change Problem (1 Viewer)

[The-Noodle]

I can't seem to get the answer for this. I can do part a) but not b). Any help is appreciated.

Sand falling from a funnel forms a conical pile such that the height of the pile is one and a half times the radius.
a) Show that the volume of the pile is given by V = πr^3/2
b) If the sand is falling at the rate of π/10 m^3 per minute, find the rate at which the height is increasing when the pile of sand is 3 m high.

 

[pLuvia]

You want dh/dt
So,
dh/dt=dh/dV*dV/dt

Now you have V=pir^3/2 now you want V in terms of h, so sub in a value so you get this answer then differentiate that do get your dV/dh, the reciprocal is what you want.

You are given that dV/dt=pi/10

So now you can find dh/dt, by subbing in h=3 into the dh/dV then multiplying that by dV/dt

 

[alcalder]

I can't seem to get the answer for this. I can do part a) but not b). Any help is appreciated.

Sand falling from a funnel forms a conical pile such that the height of the pile is one and a half times the radius.
a) Show that the volume of the pile is given by V = πr^3/2
b) If the sand is falling at the rate of π/10 m^3 per minute, find the rate at which the height is increasing when the pile of sand is 3 m high.

b) The sand falling at a certain rate... when they say rate, they mean the time is changing. SO...

dV/dt = π/10 m³min^-1

Now we also know that the height, h, of the pile is 3/2r

h=3r/2
r = 2h/3
SO
V = πr³ /2
V = 8πh³/27

We need to find dh/dt when h = 3m

dh/dt = dh/dV x dV/dt

dV/dh = 8πh²/9

dh/dV = 9/(8πh²)

dh/dt = dh/dV x dV/dt
= 9/(8πh²) x π/10
= 9/(80h²)

When h = 3
dh/dt = 1/80 = 0.0125 ms^-1

 

[SnapX]

I acctually remember doing this question, studyin for 1/2 yearlies
Part a)
h=1.5r (given in the question)
V=1/3pir^2h (volume of a cone formula)
V=1/3pir^2 * 1.5r (subbing in a value for h)
V=pi/2 * r^3

Part B (note this may be the long way around but its the way i got it)
h = 1.5r V=pi/2 * r ^2
r = h/1.5 dv/dr = 3/2 * pi * r^2
dr/dh = 2/3 at h = 3 r = 2 (from h=1.5r)
dv/dr = 3/2 * pi * 4
dv/dr = 6 * pi
dV/dt = dh/dt * dV/dh
But
DV/dh = dr/dh * dv/dr

Therefore:
pi/10 = dh/dt * 2/3 * 6 * pi
pi/10 = dh/dt * 4 * pi
dh/dt = 1/40 = 0.025

The End!