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.
b) find lim p(t) as t→∞
lim p(t) = 45% (simplify your answer.)
c) sketch a graph of p. choose the correct graph below.

Explanation:

Step1: Recall the formula for $P(t)$

Given $P(t)=Q+(100 - Q)e^{-kt}$, 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 $t
ightarrow\infty$. Then $\lim_{t
ightarrow\infty}P(t)=45+(100 - 45)\times0=45$.

Step3: Analyze the graph of $P(t)$

The function $P(t)=45 + 55e^{-0.7t}$. When $t = 0$, $P(0)=45+55e^{0}=45 + 55=100$. As $t$ increases, $P(t)$ decreases exponentially towards the horizontal - asymptote $y = 45$. The correct graph is a decreasing exponential - type function that starts at $P(0)=100$ and approaches $y = 45$ as $t
ightarrow\infty$.

Answer:

b) $\lim_{t
ightarrow\infty}P(t)=45$
c) The correct graph is a decreasing exponential - type function that starts at $(0,100)$ and approaches the horizontal asymptote $y = 45$ as $t$ increases. (Without seeing the actual graph options clearly, the description of the correct graph is given. If you can provide more details about the graph options, a more specific answer can be given).