Questions: 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=50 and k=0.6. Complete parts (a) through (e) below. 6 weeks 51.4 % 10 weeks 50.1 % (Round to one decimal place as needed.) b) Find lim t -> infinity P(t). lim t -> infinity P(t)=50% (Simplify your answer.) c) Sketch a graph of P. Choose the correct graph below. A. B. C. D.

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=50 and k=0.6. Complete parts (a) through (e) below.

6 weeks 51.4 %
10 weeks 50.1 %
(Round to one decimal place as needed.)

b) Find lim t -> infinity P(t).
lim t -> infinity P(t)=50% (Simplify your answer.)

c) Sketch a graph of P. Choose the correct graph below.
A.
B.
C.
D.

Transcript text: 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 $\mathrm{P}(\mathrm{t})=\mathrm{Q}+(100-\mathrm{Q}) e^{-k t}$, where Q is the percentage that we would never forget and $k$ is a constant that depends on the knowledge learned. Suppose that $\mathrm{Q}=50$ and $\mathrm{k}=0.6$. Complete parts (a) through (e) below. \begin{tabular}{|r|l|} \hline 6 weeks & $51.4 \%$ \\ \hline 10 weeks & $50.1 \%$ \\ \hline \end{tabular} (Round to one decimal place as needed.) b) Find $\lim _{t \rightarrow \infty} P(t)$. $\lim _{t \rightarrow \infty} P(t)=50 \%$ (Simplify your answer.) c) Sketch a graph of $P$. Choose the correct graph below. A. B. C. D.

Solution

Solution Steps

Step 1: Find the limit of P(t) as t approaches infinity.

Given the function $P(t) = Q + (100 - Q)e^{-kt}$, we want to find $\lim_{t\to\infty} P(t)$. We are given $Q = 50$ and $k = 0.6$. Substituting these values into the equation, we have $P(t) = 50 + (100 - 50)e^{-0.6t} = 50 + 50e^{-0.6t}$. As $t$ approaches infinity, $e^{-0.6t}$ approaches 0. Therefore, $\lim_{t\to\infty} P(t) = 50 + 50(0) = 50$.

Step 2: Sketch the graph of P(t)

We know that $P(0) = 50 + 50e^{-0.6(0)} = 50 + 50(1) = 100$. Also, we know that as $t$ approaches infinity, $P(t)$ approaches 50. The graph starts at 100 and exponentially decays towards 50.

Step 3: Choose the correct graph

Based on the behavior of the function derived in the previous step, the correct graph is the one that starts at P=100 when t=0 and decreases towards P=50 as t increases. This corresponds to graph B.