How fast are the a) height and b) radius changing when the pile is 4m high?

Question

Sand falls from a conveyor belt at the rate of 10m^3/min onto the top of a conical pile.The height of the pile is always three-eights of the base diameter. How fast are the a) height and b) radius changing when the pile is 4m high? Answer in centimeters per minute.

The answer of the "a" must be 11.19 cm/min.

The answer of the "b" must be 14.92 cm/min.

Thank you so much!

Answer 1

Ok, here's how you do it:

First, you know from a diagram of the problem and from the problem itself taht the height of the pile will be 3/8 of the diameter, so

h = 3/8D

it's really convinient to turn the diameter to radius now

you know that D = 2r so

h = 3/8(2r) = 3r/4

now  you have to get the volume of an ordinary cone. Since i don't remember it i will get it the old fashioned way.

It is said that the volume of a solid in revolutions will equal the area of one half of the volume in 2d multiplied by the circumference of a point located at the centroid of this figure measured from the center of rotation of the figure. In this case the half of it will be a triangle with the radius as the base and the height as it's height, so:

A=hr*2

and the centroid will be located at one third of the axis of rotation wich means that the circumverence will be

C=2*pi*r/3

multiplying

V = h*r^2*pi/3

since r = 4h/3 then

V = 16*h^3*pi/27

now, derive for h

dV= 16*h^2*pi/9 dh

now divide both sides per dt you get that

dV/dt = 16*h^2*pi/9 dh/dt

solve for dh/dt

dh/dt = 9*(dv/dt)/(16*h^2*pi)

you know that dv/dt will be 10 m^3/min and that h will be 4m, so

dh/dt = 9*(10m^3/min)/(16*(4m)^2*pi)

a) dh/dt = 11.19cm/min

now, for b you get the volume formula

V = h*r^2*pi/3

and substitute h=3r/4 so

V=r^3*pi/4

deriving for r

dV = 3*r^2*pi/4 dr

dividing into dt

dV/dt = 3*r^2*pi/4 dr/dt

solving for dr/dt

dr/dt = 4*(dV/dt)/(3*r^2*pi)

substituting r = 4h/3

dr/dt = 4*(10m^3/min)/(3*(16/3m)^2*pi)

dr/dt = 14.92 cm

hope this helps

Answer 2

the volume of a cone is:

(1). V = (h/3) x pi r^2

But the relationship of h and r is given as:

h = (3/8)(2r) = 3r/4, or r = 4h/3 thus:

(2). V = (3r/12) pi r^2 = (pi r^3)/4

dV/dr = (3 pi r^2) /4

dV = ((3pi r^2)/4)dr

but dV/dt = 10, hence;

((3pi r^2)/4) dr /dt = 10

dr/dt = 40/(3 pi r^2)

if h = 4m, r = 16/3, thus:

b). dr/dt = 40/(3 pi (16/3)^2) = 0.1492 m/min

= 14.92cm/minute

but r = (4/3) h, hence;

dr = (4/3)dh, thus;

(4/3)dh/dt = 14.92

a). dh/dt = 14.92 x3/4 = 11.19 cm/min