Problem 82 - Algebra and Trigonometry

Question

Learning Curve A learning curve is a graph of a function \(P(t)\) that measures the performance of someone learning a skill as a function of the training time \(t\) . At first, the rate of learning is rapid. Then, as performance increases and approaches a maximal value \(M,\) the rate of learning decreases. It has been found that the function $$ P(t)=M-C e^{-k t} $$ where \(k\) and \(C\) are positive constants and \(C

Step-by-Step Solution

Answer

(a) \( t = -\frac{1}{k} \ln\left(\frac{M - P}{C}\right) \). (b) 58.53 months. (c) Graph the curve: \( P(t) = 20 - 14e^{-0.024t} \).

Step 1: Express Learning Time in Terms of Performance Level

Start with the equation given: \[ P(t) = M - Ce^{-kt} \]To express \(t\) as a function of \(P\), solve for \(t\):\[ P = M - Ce^{-kt} \]Rearrange this to solve for \(e^{-kt}\):\[ e^{-kt} = \frac{M - P}{C} \]Now, take the natural logarithm on both sides:\[ -kt = \ln\left(\frac{M - P}{C}\right) \]Solve for \(t\):\[ t = -\frac{1}{k} \ln\left(\frac{M - P}{C}\right) \]

Step 2: Solve for Time to Achieve 12 ft Performance

Given the learning curve:\[ P(t) = 20 - 14e^{-0.024t} \]and you need \(P = 12\) ft.Plug \(P = 12\) into the equation:\[ 12 = 20 - 14e^{-0.024t} \]Rearrange to solve for \(e^{-0.024t}\):\[ 14e^{-0.024t} = 8 \]\[ e^{-0.024t} = \frac{8}{14} = \frac{4}{7} \]Take the natural logarithm of both sides:\[ -0.024t = \ln\left(\frac{4}{7}\right) \]Solve for \(t\):\[ t = -\frac{1}{0.024}\ln\left(\frac{4}{7}\right) \]Using a calculator:\[ t \approx 58.53 \] months.

Step 3: Graph the Learning Curve

Create a graph with \(t\) on the x-axis (time in months) and \(P(t)\) on the y-axis (performance in feet). The function is:\[ P(t) = 20 - 14e^{-0.024t} \]This graph starts high, then approaches the value \(M=20\) asymptotically. Begin by plotting points for selected \(t\) values, such as \(t = 0, 20, 40, 60, 80\), and draw a smooth curve connecting these points.

Key Concepts

Performance Function

A performance function is a mathematical representation used to describe how someone improves at a skill over time. In the context of a learning curve, which we are discussing here, the performance function often takes a specific form. For example, with the learning model \[P(t) = M - Ce^{-kt}\]M represents the maximum performance level, and both C and k are constants that influence the shape and rate of the curve. With regular practice and training, the learner's performance starts improving rapidly but slows down as they get closer to their maximum potential. This function effectively models such a process, giving us valuable insights into how fast or slowly performance might improve over time.

Exponential Decay

Exponential decay is a kind of change where a quantity diminishes at a rate proportional to its current value. In the learning curve equation, \[ Ce^{-kt} \]is the part that reflects exponential decay. The term \(e^{-kt}\)describes how fast the impact of additional learning time decreases as training continues. This impacts how quickly a learner approaches their maximum potential performance. At first, a trainee sees rapid improvement, but as time goes by, the same amount of training leads to smaller and smaller gains. This naturally aligns with real-world learning experiences where initial gains are more noticeable than later ones.

Asymptotic Behavior

Asymptotic behavior refers to how a function behaves as it approaches a certain value, often at the extremes of its domain. For the learning curve, the function \[P(t) = 20 - 14e^{-0.024t}\]demonstrates asymptotic behavior towards the maximum performance level of 20. As training time increases, the learner's performance never quite hits this maximum value but gets closer and closer. This indicates that full mastery or ultimate skill level is approached but technical perfection remains elusive, highlighting a practical aspect of learning where improvements continue to taper off over extensive periods of practice.

Pole Vaulting Training

Pole vaulting training aims to improve an athlete's ability to clear certain heights. When using a performance function like \[P(t) = 20 - 14e^{-0.024t}\],it models the pole vaulter's learning curve over time. Here, \(P(t)\) represents the height the athlete can achieve, measured in feet after \(t\) months. The model shows that after approximately 58.53 months, the athlete can clear 12 feet, given the initial conditions set. Understanding this model helps coaches and athletes set realistic training goals and expectations, focusing on areas of improvement that are crucial during early stages when the ability to clear new heights rapidly increases.