Question

The model Upper P equals StartFraction 0.9 x minus 0.3 Over 0.9 x plus 0.6 EndFraction
is a learning​ curve, where P is the proportion of correct responses after x trials. If Pequals​0, there are no correct responses. If Pequals​1, all responses are correct. The graph of the rational formula is shown below. How many learning trials are necessary for 0.94 of the responses to be​ correct? Identify your solution as a point on the graph.
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Number of Learning Trials
Proportion of Correct Responses



A coordinate system has a horizontal x-axis labeled Number of Learning Trials from 0 to 20 in increments of 1 and a vertical P-axis labeled Proportion of Correct Responses from 0 to 1.2 in increments of 0.1. From left to right, a curve rises steeply at first, passing through point (1, 0.4), then rises at a decreasing rate passing through point (5, 0.82) before approaching an asymptote at P = 1. All coordinates are approximate.
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Question content area right
Part 1
  
1 learning trials

Answer 1

To find how many learning trials are necessary for the proportion of correct responses \( P \) to be 0.94, we need to solve the equation given by the learning curve model:

\[
P = \frac{0.9x - 0.3}{0.9x + 0.6}
\]

Setting \( P = 0.94 \):

\[
0.94 = \frac{0.9x - 0.3}{0.9x + 0.6}
\]

To eliminate the fraction, we can cross-multiply:

\[
0.94(0.9x + 0.6) = 0.9x - 0.3
\]

Expanding both sides, we get:

\[
0.846x + 0.564 = 0.9x - 0.3
\]

Now, let's rearrange the equation to isolate \( x \):

\[
0.846x + 0.564 + 0.3 = 0.9x
\]
\[
0.846x + 0.864 = 0.9x
\]

Subtracting \( 0.846x \) from both sides:

\[
0.864 = 0.9x - 0.846x
\]
\[
0.864 = 0.054x
\]

Now, divide both sides by \( 0.054 \):

\[
x = \frac{0.864}{0.054} \approx 16
\]

Thus, approximately **16 learning trials are necessary** for the proportion of correct responses to reach 0.94.

As a point on the graph, this is represented as:

\[
(16, 0.94)
\]