Question

CALCULUS DERIVATIVE help: At a sand and gravel plant, sand is falling off a conveyor and onto a conical pile at a rate of 10 cubi feet per minute. The diameter of the base of the one is approx three times the altitude At what rate is the height of the pile changing when the pile is 15 feet high?

Answer 1

Ok so basically, I'm looking for the derivative of the height right?

Answer 2

So I use the formula to find the volume of a cone?

Answer 3

Rate of Change. Correct?

Answer 4

yes, of the height.

Answer 5

You will need an expression for the volume of a cone and will have to use the chain rule to obtain a relationship between dv/dt and dh/dt

Answer 6

ok so v=pi*r^2*h/3

Answer 7

dv/dt=pi*2r*dr/dt* -3h^-2 * dh/dt

Answer 8

Your equation for Volume is correct but the derivative is not.
Also note the problem told you something about the diameter of the base of the cone.
You will need to use that...do you see why?

Answer 9

oooooh ok

Answer 10

so the diameter is 3x the height?

Answer 11

V = 1/3*pi*r^2*h agreed

Answer 12

Now the diameter is 1/3 of the height...but the Volume formula does not have the diameter in it...how do you introduce it?

Answer 13

2*r

Answer 14

ok so r = d/2 and d=h/3 substitute these into your formaula for the Volume and lets see what you get

Answer 15

2r=3h
r=?

Answer 16

r=3h/2

Answer 17

So what does the Volume formula simplify to?

Answer 18

but wait, I'm confused now...hmmm after substituting I get v= 1/3*pi*3h/2

Answer 19

Do it one step at a time.

Answer 20

\[V=\frac{ 1 }{ 3 } \pi r^2 h=\frac{ \pi }{ 3 }\left( \frac{ 3h }{ 2 } \right)^2h=\frac{ 3 \pi }{ 4 }h^3\]
dV/dt=?

Answer 21

ok so the volume of a cone is π*r^2*h/3
The diameter is 3x the height.
and radius is d/2

Answer 22

So V = what?

Answer 23

but how do you get 3h/2? because After plugging in h/3 in r=d/2, I get 2h/3 instead..

Answer 24

Look at what surithayer got...do you get the same?

Answer 25

My mistake d = 3h

Answer 26

yeah because it's 3 times

Answer 27

ok

Answer 28

and where did the 1/3 come from?

Answer 29

that was my mistake ...sorry.

Answer 30

the 1/3 ?

Answer 31

When I google the formula for the volume of a cone I get v=pi*r^2*h/3

Answer 32

ok
V = 1/3 pi* r^2 * h that is the volume of a cone...so the 1/3 comes from this formula

Answer 33

yes..see the 1/3 at the end?

Answer 34

I thought the 3 is only under the h

Answer 35

So V = 1/3*pi*r^2*h which is the same as pi*r^2*h/3

Answer 36

oh ok :)

Answer 37

So now that v=3pi/4 *h^3, can I take the derivative?

Answer 38

Yes...but take it carefully using the chain rule. Lets see what you get.

Answer 39

dv/dt = 3pi/4 *3h^2 *dh/dt

Answer 40

Yes...now substitute and solve for dh/dt

Answer 41

ok ^_^

Answer 42

Get an answer?

Answer 43

Ok so dh/dt = 8/405 pi :D

Answer 44

yay thanks so much ^_^

Answer 45

I got .00629 ft/min wheras your answer works out to .062...just double check

Answer 46

your welcome ...