F.5 maths m1 differentiation

t is because the rate of change (melt) is depend on time. However, the volume of the ice block is decreasing while the time increase. Therefore, the rate of the ice block melts is a negative value.

The definition of rate change:
The rate change depends on time.
Therefore,
The volume change by the time:V/t
However, there is a very little time that we cannot calculate.
For example, if the function of the volume changing of the ice block is V(t):t^2
Change in volume:(t+Δt)^2 - t^2 (which Δt is a very short period of time)
Average change in volume:[(t+Δt)^2 - t^2]/Δt (The average change in volume in a Δt of time)
The following substitution:
Time interval | Change in volume | Average change in volume
t=2 to t=2.5 | 2.5^2 - 2^2 | (2.5^2 - 2^2)/(2.5-2)=4.5
t=2 to t=2.1 | 2.1^2 - 2^2 | (2.1^2 - 2^2)/(2.1-2)=4.1
t=2 to t=2.001 | 2.001^2 - 2^2 | (2.001^2 - 2^2)/(2.001-2)=4.001
Hence, when the Δt get smaller and smaller, the average change in volume gets closer and closer to the instantaneous rate of change in volume. (The change in a very short moment of time)

Therefore,
dV/dt =
lim [V(t+Δt) - V(t)]/Δt
(Δt -> 0)
In above function, we can get that if the time is 2 minutes,
V(t)=t^2
dV/dt= 2t
dV/dt(2) = 2(2) = 4cm^3/min

Hence, the above result show that when the time is 2 minutes (time get closer to 2 minutes), the rate of change in volume respect to time is 4cm^3/min. (the change of volume get closer to 4cm^3 at the time on 2 minutes)