Question

can you explain to me how to do this question i know i have to use differential equations but im not sure how to form an equation for inversely proportional

question states

grain is being poured at a steady rate to form a pile with height h, the rate at which the height is increasing is inversely proportional to h^3.
the initial height of the pile is ho and the height doubles after time T.

find in terms of T, the time after which the height of the pile is 3ho

if you could explain it as much as possible

thanks

Answer 1

dh/dt = k/h^3 where k is unknown constant
h^3 dh = k dt
h^4/4 + c = kt
at t = 0, h = ho
ho^4/4 + c = 0
c = -ho^4/4
so
h^4/4 -ho^4/4 = k t
at t = T, h = 2 ho
16 ho^4/4 -ho^4/4 = k T
15 ho^4/4 = k T
k = (1/T)(15 ho^4/4)
so
h^4 - ho^4 = 25 ho^4 (t/T)
when is h = 3 ho ?
81 ho^4 - ho^4 = 25 ho^4 (t/T)
80 = 25 (t/T)
t = 80T/25 = 16 T/5

Answer 2

i follow you up to this point

h^4 - ho^4 = 25 ho^4 (t/T)
when is h = 3 ho ?
81 ho^4 - ho^4 = 25 ho^4 (t/T)
80 = 25 (t/T)
t = 80T/25 = 16 T/5

where do you get 25 from and 81