rates of change question (1 Viewer)

[kooltrainer]

The answer 7b) is 32000Pi/3 cm cube.
i just need help with 7b)

 

[random-1005]

The answer 7b) is 32000Pi/3 cm cube.
i just need help with 7b)

dv/dt= dv/dr * dr/da * da/dt (apply chain rule multiple times, easy way to check differentials behave like fractions, notice it will cancel down to dv/dt)

v= (4/3pi)r^3
dv/dr= 4pi * r^2

a= 4pi r^2
da/dr= 8 pi r ---> dr/da= 1/ (8 pi r)

given da/dt = +1


dv/dt= dv/dr * dr/da * da/dt



dv/dt= r/2= 10

r= 20

V= (4/3 pi ) (20^3) cm ^3---> evaluate

 

[kooltrainer]

what i did was :
dv/dt = dv/da *da/dt
since v = 4/3 * Pi r^3, a=4Pi*r^2, so v = (1/3)*r*a, dv/da=(1/3)*r

so 10 = (1/3)*r* 1
r = 30..
as opposed to
r=20 in ur case..

why is my case wrong?

 

[random-1005]

what i did was :
dv/dt = dv/da *da/dt
since v = 4/3 * Pi r^3, a=4Pi*r^2, so v = (1/3)*r*a, dv/da=(1/3)*r

so 10 = (1/3)*r* 1
r = 30..
as opposed to
r=20 in ur case..

why is my case wrong?

\

im not very sure, one suspects its this step, im pretty sure you cannot say area is a constant, you would use implicit differentiation maybe there, not 100% sure, but the whole method of expressing volume in terms of area looks dodgy

 

[Affinity]

what i did was :
dv/dt = dv/da *da/dt
since v = 4/3 * Pi r^3, a=4Pi*r^2, so v = (1/3)*r*a, dv/da=(1/3)*r

so 10 = (1/3)*r* 1
r = 30..
as opposed to
r=20 in ur case..

why is my case wrong?

This: dv/da=(1/3)*r,
the problem here is r is not a constant with respect to a.



correct statement is : dv/da = 1/2 r
v = a * r/3
v = a * 1/3 sqrt(a/4pi)
v =a^(3/2) * 1/3 sqrt(1/4pi)
dv/da = 1/2 sqrt(a) *1/sqrt(4pi)
= r/2

alternatively

v = a* r/3

dv/da = {da/da } * r/3 + a * d/da [r/3]
= r/3 + a/3 * dr/da
you know that da/dr = 8pi* r
and that dr/da * da/dr = 1 by the chain rule. so
dv/da = r/3 + a/3 * (1/8pi*r) = r/3 + r/6 = r/2

 

[random-1005]

This: dv/da=(1/3)*r

lol, moral of the story, keep it simple stupid

 

[kooltrainer]

shit, this makes a lot of sense.. r isnt constant wrt v ..ty