Posted by Ariana on Saturday, August 16, 2014 at 4:11am.
for any height h, volume is 1/3 PI r^2 h
Notice your formula is wrong.
but for r for any height h is r=h/2. Think on that.
so volume=1/3 PI *(h/2)^2*h
V=1/12 * PI * h^3
and h= cuberoot (12V/PI) solve for h when V=1/8*1/12*PI*2 (max volume, r=1,h=2)
Now for the calculus work.
dV/dh=3/12 PI h^2
but dV/dh*dh/dt=dV/dt
or dh/dt=dV/dt / dV/dh
you area given dV/dt as PI m^3/minute
and you found dV/dh=3/12 PI h^2
so figure dh/dt
I understand the rest but not the 1/8th of the volume part. Can you give me a further explanation on that?
Thanks :)
Answer is dh/dt= m/min
The volume when it's 1/8th full is Pi/12
I don't understand how they got that....
make a sketch to see that by ratios,
r = h/2 , like bobpursely noted
The sneaky part of the question is that when the cone is 1/8 full , the water is NOT 1/8 of the way up
Full volume = (1/3) π (1^2)(2) = 2π/3 m^3
we want r and h when volume = (1/8)(2π/3) or π/12
(Again, see bobpursely above)
Volume = V = (1/3)π(r^2)(h)
= (1/3)π(h^2/4)(h) = (π/12) h^3
so when cone is 1/8 full,
(π/12) h^3 = π/12
h^3 = 1
h = 1 , and r = 1/2
Now back to actual Calculus,
V = (π/12 h^3
dV/dh = (π/4) h^2 dh/dt
plug in the given dV/dt = π, and h = 1
π = (π/4) (1^2) dh/dt
dh/dt = 1/4
So when the cone is 1/8 full, the height is changing at
1/4 m/minute
We could have done the 1/8 part in our heads by realizing that ..
The volume of two similar solids is proportional to the cube of their sides, and since
(1/2)^3 = 1/8 ......
the height must have been 1/2 of the 2 m of the cone, or 1 m