Question

Sand is being emptied from conveyor belt onto the ground.

The sand is piling up in the shape of a cone.

Sand is currently being added to the pile at a rate of 50ft3 / sec.

At what rate is the height of the pile increasing when there is 1200ft 3 of sand, and the radius is 10 feet and increasing at 0.15 ft/sec?

Answer 1

1200 cubic feet = volume of sand
radius of cone = 10
rate of increase of radius = \(\Large{dr\over dt}\)=0.15
rate of volume increase = \(\Large {dV\over dt}\)=50

Answer 2

\[V={1\over 3}\pi r^2h\\
{dV\over dt}=\left({1\over3}\pi\times2r{dr\over dt}\right)h+\left({1\over3}\pi r^2\right){dh\over dt}
\]

Answer 3

kapeesh?

Answer 4

yes

Answer 5

did you follow the derivative?

Answer 6

\[(\frac{ 1 }{ 3}\pi \times2r0.15)h+(\frac{ 1 }{ 3}\pi r^2)\frac{ dh }{ dt }\]yeah I got

Answer 7

great.. we are asked to find \(\Large{dh\over dt}\).. plug in the numbers that I extracted from the given information

Answer 8

you are missing the \[dV\over dt\] on the left side.. we need it

Answer 9

50= then what I wrote

Answer 10

yep..

Answer 11

ok I got that part what do I do next

Answer 12

plug in the numbers and solve for \[dh\over dt\]

Answer 13

ok but what does r=

Answer 14

r = radius of the cone

Answer 15

all the values are up there except for one

Answer 16

ok what does h=

Answer 17

exactly
\[V={1\over3}\pi r^2h\\
1200={1\over3}\pi(10)^2h\]solve for h

Answer 18

ok my calculator is not letting me calculate can yo help me calculate\[\frac{ 1 }{ 3 }\pi x 2 (10)(0.15)11.46\frac{ 1 }{ 3 }\pi (10)^2\]

Answer 19

try the wolframalpha.com

Answer 20

wait I thi nk I got it

Answer 21

but check with the +, - signs

Answer 22

great

Answer 23

ok I got\[\frac{ dh }{ dt }=.01326\]
is that right

Answer 24

well, I did not calculate myself but thats how you gotta do it
:)

Answer 25

is that it or is there more

Answer 26

well, that is all that the question is askin, aint it

Answer 27

yess I was just making sure

Answer 28

lets not give more than what they ask for

Answer 29

thanks for all your help

Answer 30

yer welcome.