Problem 66 - A Graphical Approach to Precalculus with Limits

Question

During the 100 -meter dash, the elapsed time \(T\) for a sprinter to reach a speed of \(x\) meters per second is given by the following function. $$T(x)=-1.2 \ln \left(1-\frac{x}{11}\right)$$ (a) How much time had elapsed when the sprinter was running 0 meters per second? Interpret your answer. (b) At the end of the race, the sprinter was moving at 10.998 meters per second. What was the sprinter's time for this 100 -meter dash? (c) Find \(T^{-1}(x)\) and interpret its meaning.

Step-by-Step Solution

Answer

(a) 0 seconds; no movement yet. (b) 10.428 seconds. (c) \(T^{-1}(x) = 11(1 - e^{-\frac{x}{1.2}})\); it provides the speed after \(x\) seconds.

Step 1: Finding elapsed time for 0 m/s

To find the time when the sprinter is running at 0 meters per second, substitute \(x = 0\) into the function \(T(x)\). Calculate:\[T(0) = -1.2 \ln \left(1 - \frac{0}{11}\right) = -1.2 \ln(1)\]Since \(\ln(1) = 0\), we have:\[T(0) = 0\]This means no time has elapsed as the sprinter is not yet moving.

Step 2: Determining time for a speed of 10.998 m/s

Substitute \(x = 10.998\) into the function. Calculate:\[T(10.998) = -1.2 \ln \left(1 - \frac{10.998}{11}\right) = -1.2 \ln(1 - 0.9998181818)\]\[T(10.998) = -1.2 \ln(0.0001818182)\]Evaluate the natural logarithm and then multiply:\[\ln(0.0001818182) \approx -8.69\]\[T(10.998) = -1.2 \times -8.69 \approx 10.428\]The sprinter's time for this 100-meter dash is approximately 10.428 seconds.

Step 3: Finding the inverse function \(T^{-1}(x)\)

To find the inverse function \(T^{-1}(x)\), switch the roles of \(x\) and \(T\) in the original equation and solve for \(x\):\[x = -1.2 \ln \left(1 - \frac{T}{11}\right)\]Rearrange to solve for \(T\):\[-\frac{x}{1.2} = \ln \left(1 - \frac{T}{11}\right)\]Exponentiate both sides:\[e^{-\frac{x}{1.2}} = 1 - \frac{T}{11}\]Rearrange to isolate \(T\):\[\frac{T}{11} = 1 - e^{-\frac{x}{1.2}}\]\[T = 11 \left(1 - e^{-\frac{x}{1.2}}\right)\]The inverse function \(T^{-1}(x)\) tells us the speed achieved in \(x\) seconds.

Key Concepts

Inverse Functions

Understanding inverse functions can be a bit tricky, but they're incredibly useful. Essentially, an inverse function reverses the operation of the original function. It takes the output value and produces the input value that generated it. For instance, in our scenario of the 100-meter dash, the function \( T(x) \) gives us the time it takes for a sprinter to reach a certain speed \( x \). The inverse function \( T^{-1}(x) \) is useful because it tells us the speed achieved at a specific time.

When finding an inverse function, we switch the input and output. If \( T(x) \) outputs time for a given speed, \( T^{-1}(x) \) will output speed for a given time. This function helps solve problems where you need to know the performance at a certain point in time.

Natural Logarithms

Natural logarithms, represented by \( \ln \), are essential in advanced mathematical calculations. They're logarithms to the base \( e \), where \( e \) is approximately 2.718. Natural logarithms have a unique ability to simplify exponential equations, and they often appear in growth and decay problems.

In the context of the 100-meter dash, the function \( T(x) = -1.2 \ln \left(1 - \frac{x}{11}\right) \) uses natural logarithms to model elapsed time based on sprinter speed. The natural logarithm part of this equation, \( \ln \left(1 - \frac{x}{11}\right) \), helps in capturing the non-linear relationship between speed and time, accounting for the acceleration phase of a sprinter.

Elapsed Time

Elapsed time is a simple yet crucial concept in time measurement. It represents the total time that has passed from the beginning to a specific point in an event. In the 100-meter dash example, the function \( T(x) \) calculates the elapsed time based on the speed of the sprinter.

When the sprinter is at 0 meters per second, \( T(0) \) is 0, indicating that no time has passed. As the sprinter's speed increases, the elapsed time computed by \( T(x) \) grows, signifying the sprinting progress. Knowing how to calculate elapsed time allows athletes and coaches to break down performances and strategize improvements.

100-meter Dash

The 100-meter dash is one of the most popular track and field events, known for its speed and intensity. This event tests the sprinter's ability to accelerate quickly and maintain top speed over a short distance. In terms of physics, it's all about acceleration, speed, and time.

Analyzing performances of athletes in the 100-meter dash often involves calculating specific metrics like elapsed time for various speeds. The function \( T(x) \) used in this context, applies mathematical modeling to capture and predict performance. Mathematics helps coaches and athletes understand how quickly a sprinter reaches peak speed, which is crucial for optimizing performance and training.