A vessel is being filled at a variable and the volume of liquid in the vessel at any time t is given by V = A( 1- e^-kt) V
a) Show that dV/dt = k(A-V)
b) if one quarter of the vessel is filled in 5 minutes, what fraction is filled in the next 5 minutes?
c) show that lim (x-> infinite) = A.
Thank you!
a) To show that dV/dt = k(A-V), we need to take the derivative of V with respect to t. Start by rewriting V = A(1 - e^(-kt)) as V = A - Ae^(-kt). Now, differentiate both sides of the equation with respect to t using the chain rule:
dV/dt = d(A - Ae^(-kt))/dt
= dA/dt - d/dt(Ae^(-kt))
= 0 - (-Ake^(-kt)) [since dA/dt = 0 and d/dt(e^(-kt)) = -ke^(-kt)]
= kAe^(-kt)
Next, we substitute the value of V into the expression:
k(A - V) = k(A - (A - Ae^(-kt)))
= k(A - A + Ae^(-kt))
= kAe^(-kt)
Therefore, we have shown that dV/dt = k(A-V).
b) In the next 5 minutes, we need to find the fraction of the vessel that is filled. Let's denote this fraction as f. We know that one quarter of the vessel is filled in 5 minutes, so after 5 minutes, the volume V will be equal to (1/4)A.
Using the given equation V = A(1 - e^(-kt)), we can substitute the known values and solve for k:
(1/4)A = A(1 - e^(-kt))
1/4 = 1 - e^(-kt)
e^(-kt) = 1 - 1/4
e^(-kt) = 3/4
-kt = ln(3/4)
k = -ln(3/4)/t
Now we can determine the fraction of the vessel that is filled in the next 5 minutes by substituting t = 5 into the equation:
V = A(1 - e^(-kt))
V = A(1 - e^(-5(-ln(3/4))/5))
V = A(1 - e^(ln(3/4)))
V = A(1 - 3/4)
V = A/4
Therefore, in the next 5 minutes, one quarter (1/4) of the vessel will be filled.
c) To show that lim (t -> infinity) V = A, we need to take the limit as t approaches infinity of the volume equation V = A(1 - e^(-kt)).
lim (t -> infinity) V = lim (t -> infinity) A(1 - e^(-kt))
As t approaches infinity, the term e^(-kt) approaches 0, since negative exponents tend to zero as x approaches infinity:
lim (t -> infinity) e^(-kt) = 0
Therefore,
lim (t -> infinity) V = lim (t -> infinity) A(1 - 0) = A
This shows that as t approaches infinity, the volume in the vessel, V, approaches A.