Question

Calculus: Find the derivative.
Sand is falling off a conveyor belt onto a conical pile at the rate of 16 ft^3 per minute. The diameter is 4 times the height. At what rate is the height of the pile changing when h=6 feet?

Answer 1

what would 16ft^3 per minute be classified as? dv/dt?

Answer 2

yep. you can tell because the units are cubic feet (volume) per minute (time) ie dV/dt

Answer 3

so i use the same equation again? \[V=\frac{ 4 }{ 3 } \pi r ^{3} \]

Answer 4

and d=4h? rite?

Answer 5

nope... this is a cone... what's the expression for the volume of a cone?

Answer 6

\[\frac{ 1 }{ 3 }\pi r ^{2} h\]

Answer 7

and then i would find the derivative of this equation correct? :D

Answer 8

make it easier on yourself first... it says d= 4h

Answer 9

yeah i said that already :p

Answer 10

use that fact.... 2r = 4h
r=2h
plug that into the volume expression for r

Answer 11

why would 2r=4h?

Answer 12

oh wait nvm

Answer 13

k

Answer 14

ok so the derivative of this function is \[\frac{ 2 }{ 3}\pi r h\]

Answer 15

and we want to know dh/dt?

Answer 16

yes, you want to know dh/dt.

Answer 17

so is the equation now \[\frac{ dV }{ dt}= \frac{ 2 }{ 3 } \pi 2h \times \frac{ dh }{ dt }\]

Answer 18

because \[d=4h \]

Answer 19

\[2r=4h\]

Answer 20

\[r=2h\]

Answer 21

substitute r=2h

into (pi/3)*r^2*h

Answer 22

then take the derivative.

Answer 23

so its now \[V=1/3\pi4h ^{2}h\]

Answer 24

yes. \[ V = \frac{ 4\pi }{ 3 }h^3\]

Answer 25

now find
\[\frac{ d }{ dt } (V = \frac{ 4\pi }{ 3 }h^3)\]

Answer 26

so take t he derivative of that?

Answer 27

yes

Answer 28

which is \[4\pi h ^{3}\]

Answer 29

i mean 12

Answer 30

try again. remember to use the chain rule. like last time.

Answer 31

\[12\pi h ^{2}\]

Answer 32

4pih^2 *dh/dt

Answer 33

\[\frac{ dV }{ dt } =4 \pi h^2 \frac{ dh}{ dt }\]

Answer 34

wait i got \[4\pi h ^{2}\]

Answer 35

divide both sides by 4 pi h^2
plug in the given values for dV/dt and h
calculate dh/dt

Answer 36

what does h=?