The volume of grains in a silo at a particular time (measured in hours) is given
by V (t) = 4t(3-t)m^3
Find the rate of change of the volume of grains in the
silo from first principles (using the definition of the rate of change)
content appears to be that of MAS164
To find the rate of change of the volume of grains in the silo from first principles, we need to use the definition of the rate of change, which is the derivative of the function with respect to time.
The volume function is given by V(t) = 4t(3-t) m^3. We need to find the derivative dV/dt.
Let's start by applying the product rule to the function V(t) = 4t(3-t):
V(t) = 4t(3-t)
= 12t - 4t^2
Now, we can differentiate this function term by term:
dV/dt = d(12t)/dt - d(4t^2)/dt
Using the power rule of differentiation, we have:
d(12t)/dt = 12
d(4t^2)/dt = 8t
So, we can simplify the expression:
dV/dt = 12 - 8t
Therefore, the rate of change of the volume of grains in the silo, from first principles, is given by dV/dt = 12 - 8t.