Question

suppose that you are given the task of learning 100% of a block of knowledge. human nature is such that we retain only a percentage p of knowledge t weeks after we have learned it. the ebbinghaus learning model asserts that p is given by p(t)=q+(100 - q)e^{-kt}, where q is the percentage that we would never forget and k is a constant that depends on the knowledge learned. suppose that q = 45 and k = 0.7. complete parts (a) through (e) below.
a) find the percentage retained after 0 weeks, 1 week, 2 weeks, 6 weeks, and 10 weeks. complete the table to the right.
b) find \\(\lim_{t\to\infty}p(t)\\)
\\(\lim_{t\to\infty}p(t)=\square\\%\\) (simplify your answer.)

Explanation:

Step1: Recall the Ebbinghaus learning - model formula

$P(t)=Q+(100 - Q)e^{-kt}$, with $Q = 45$ and $k=0.7$.

Step2: Find $\lim_{t

ightarrow\infty}P(t)$
As $t
ightarrow\infty$, the term $e^{-kt}=e^{- 0.7t}
ightarrow0$ since the exponent $-0.7t
ightarrow-\infty$ as $t
ightarrow\infty$.
Substitute $e^{-kt}
ightarrow0$ into the formula $P(t)=Q+(100 - Q)e^{-kt}$.
We get $\lim_{t
ightarrow\infty}P(t)=Q+(100 - Q)\times0$.
Since $(100 - Q)\times0 = 0$, then $\lim_{t
ightarrow\infty}P(t)=Q$.
Given $Q = 45$, so $\lim_{t
ightarrow\infty}P(t)=45$.

Answer:

$45$