Question

In the theory of learning, the rate at which a subject is memorized is assumed to be proportional to the amount that is left to be memorized. Assume that the rate at which material is forgotten is proportional to the amount memorized. Suppose M denotes the total amount of a subject to be memorized and A(t) is the amount memorized in time t > 0. Determine a differential equation for the amount A(t) when forgetfulness is taken into account. (Assume the constants of proportionality for the rate at which material is memorized and the rate at which material is forgotten are k1 > 0 and k2 > 0, respectively. Use A for A(t).)

Answer 1

Answer: equation = dA/dt = K₁ (M - A) - K₂A

Explanation:

Let us follow this carefully to get our equation right.

From the question we are told to (Assume the constants of proportionality for the rate at which material is memorized and the rate at which material is forgotten are k1 > 0 and k2 > 0, respectively. Use A for A(t) ).

The Rate of memorizing the material is ∝ [Total amount to be memorized] - [Amount memorized]

remember using A for A(t).

where A rep the amount memorized at a time t and M is the Total amount memorized.

so we have that dA/dt ∝ M - A

which is dA/dt = K₁ (M - A) ---------------(*)

remember it was stated that forgetfulness is taken into account

Forgetfulness, F ∝ A

F = K₂A --------------- (**)

which implies the rate of memorization is decreased as a result of forgetting;

this leads to

dA/dt = K₁ (M - A) - K₂A

The rate of differential equation is ⇒ dA/dt = K₁ (M - A) - K₂A

cheers i hope this helped!!!!