Problem 106 - Calculus and its applications

Question

A keyboarder learns to type \(W\) words per minute after \(t\) weeks of practice, where \(W\) is given by $$W(t)=100\left(1-e^{-0.3 t}\right)$$. a) Find \(W(1)\) and \(W(8)\). b) Find \(W^{\prime}(t)\) c) After how many weeks will the keyboarder's speed be 95 words per minute? d) Find \(\lim _{t \rightarrow \infty} W(t),\) and discuss its meaning.

Step-by-Step Solution

Answer

After 1 week, 25.92 wpm; after 8 weeks, 90.93 wpm; derivative: \(30e^{-0.3t}\); 95 wpm after 10 weeks; speed limit is 100 wpm.

Step 1: Substitute and Calculate W(1)

To find \( W(1) \), plug \( t = 1 \) into the equation: \[ W(1) = 100 \left( 1 - e^{-0.3 \times 1} \right) \]Calculate: \[ W(1) = 100 (1 - e^{-0.3}) \approx 100 (1 - 0.7408) \approx 100 \times 0.2592 \approx 25.92 \]So, the keyboarder types approximately 25.92 words per minute after 1 week.

Step 2: Substitute and Calculate W(8)

To find \( W(8) \), plug \( t = 8 \) into the equation: \[ W(8) = 100 \left( 1 - e^{-0.3 \times 8} \right) \]Calculate: \[ W(8) = 100 (1 - e^{-2.4}) \approx 100 (1 - 0.0907) \approx 100 \times 0.9093 \approx 90.93 \]So, the keyboarder types approximately 90.93 words per minute after 8 weeks.

Step 3: Find the Derivative W'(t)

To find the derivative \( W'(t) \), apply the chain rule:\[ W(t) = 100 \left( 1 - e^{-0.3t} \right) \]The derivative is \[ W'(t) = 100 \times (-1) \times (-0.3) \times e^{-0.3t} = 30 e^{-0.3t} \].

Step 4: Solve for 95 Words Per Minute

Set \( W(t) = 95 \) and solve for \( t \):\[ 95 = 100 \left( 1 - e^{-0.3t} \right) \]\[ 0.95 = 1 - e^{-0.3t} \]\[ e^{-0.3t} = 0.05 \]Take the natural logarithm of both sides:\[ -0.3t = \ln(0.05) \]\[ t = \frac{\ln(0.05)}{-0.3} \approx \frac{-2.9957}{-0.3} \approx 9.99 \]The keyboarder will type 95 words per minute after approximately 10 weeks.

Step 5: Compute the Limit of W(t) as t Approaches Infinity

Evaluate the limit as \( t \to \infty \):\[ \lim _{t \to \infty} W(t) = \lim _{t \to \infty} 100(1 - e^{-0.3t}) \]As \( t \to \infty \), \( e^{-0.3t} \to 0 \), hence\[ \lim _{t \to \infty} W(t) = 100(1 - 0) = 100 \]This means that the maximum typing speed the keyboarder approaches is 100 words per minute.

Key Concepts

Exponential Functions

Exponential functions are mathematical functions of the form \( f(t) = a \cdot e^{bt} \), where \( e \) is the base of the natural logarithms, approximately equal to 2.718. These functions are used to model growth or decay processes. In this context, we have an equation for words typed per minute, \( W(t) = 100(1 - e^{-0.3t}) \). The negative exponent \(-0.3t\) suggests that the function models a process of diminishing returns over time, reflecting a decrease in the rate at which additional words per minute are gained with practice.

Exponential functions are commonly applied in scenarios like population growth, radioactive decay, and learning curves. In this case, the keyboarder's typing skill increases exponentially towards a maximum limit, which is determined by the parameters of the function. The use of exponential functions allows us to predict future performance and understand the pace of improvement.

Differentiation

Differentiation is a key concept in calculus that deals with finding the rate at which a function changes. It is often denoted as \( f'(t) \) or \( \frac{df}{dt} \). In this problem, to find how quickly the keyboarder's typing speed is improving, we take the derivative of \( W(t) \) with respect to \( t \).

Using the chain rule, the derivative is computed as:

• \( W(t) = 100(1 - e^{-0.3t}) \)
• \( W'(t) = 30 e^{-0.3t} \)

This equation describes the rate of change of typing speed over time. Initially, when \( t \) is small, the rate of change is higher, meaning the keyboarder learns quickly. However, as \( t \) increases, the derivative decreases because the term \( e^{-0.3t} \) approaches zero. Hence, the learning rate slows down over time, indicating diminishing returns.

Limits

Limits help us understand the behavior of functions as they approach a certain point or infinity. In the keyboarder exercise, we need to evaluate the limit of \( W(t) \) as \( t \to \infty \). This means we are interested in what happens to the word typing rate after a very long period of practice.

As \( t \to \infty \), the term \( e^{-0.3t} \) approaches zero, making

• \( \lim_{t \to \infty} W(t) = 100(1 - 0) = 100 \)

This limit signifies that the typing speed will approach a maximum of 100 words per minute but will never exceed it. The concept of a limit is crucial in calculus because it helps us identify asymptotic behavior and long-term trends in mathematical models, providing insights into both potential and limitations of a process.

Word Problems

Word problems are practical applications of mathematical concepts in real-world scenarios. They help in transferring academic knowledge to everyday settings, thus improving problem-solving skills. In this exercise, the word problem involves using exponential functions to model the skill development of a keyboarder.

The task includes evaluating specific instances \( (W(1), W(8)) \), determining a rate \( (W'(t)) \), and predicting future outcomes \( (W(t) = 95) \). These typical word problem steps require:

• Substituting values and calculating results, which involves understanding how to work with equations.
• Solving equations to find unknowns such as time, which provides insights into how long it takes to reach specific skill levels.
• Interpreting the results in the context of the problem, explaining what calculated values mean in practical terms.

Word problems ground abstract math concepts in reality, making them valuable for developing logical and analytical skills in students.