Problem 51 - Calculus Volume 1

Question

The tortoise versus the hare: The speed of the hare is given by the sinusoidal function \(H(t)=1-\cos ((\pi t) / 2)\) whereas the speed of the tortoise is \(T(t)=(1 / 2) \tan ^{-1}(t / 4), \quad\) where \(t\) is time measured in hours and the speed is measured in miles per hour. Find the area between the curves from time \(t=0\) to the first time after one hour when the tortoise and hare are traveling at the same speed. What does it represent? Use a calculator to determine the intersection points, if necessary, accurate to three decimal places.

Step-by-Step Solution

Answer

Find the integral of the difference between the speeds from \(t=0\) to the first intersection point after one hour.

Step 1: Graph the Functions

First, we need to plot the functions \(H(t) = 1 - \cos\left(\frac{\pi t}{2}\right)\) and \(T(t) = \frac{1}{2}\tan^{-1}\left(\frac{t}{4}\right)\) to understand their behavior over time. This will help us identify the intersection points where the speeds of the hare and tortoise are equal.

Step 2: Find the Intersection Points

Set \(H(t) = T(t)\) and solve the equation \(1 - \cos\left(\frac{\pi t}{2}\right) = \frac{1}{2}\tan^{-1}\left(\frac{t}{4}\right)\) to find the values of \(t\) where the speeds are equal. Use a calculator to find the intersection point accurate to three decimal places.

Step 3: Determine Intersection Time after One Hour

The objective is to find the first time after \(t = 1\) hour where the speeds are equal. Using numerical tools or a graphing calculator, determine this intersection point within suitable limits of \(t\) beyond 1 hour.

Step 4: Setup the Integral for the Area Between Curves

The area between the curves from \(t = 0\) to the intersection point \(t = a\) can be determined by the definite integral \(\int_{0}^{a} \left| H(t) - T(t) \right| \, dt\). Calculate \(a\) based on the intersection found in the previous step.

Step 5: Integrate to Find the Area

Evaluate the integral \(\int_{0}^{a} \left(1 - \cos\left(\frac{\pi t}{2}\right) - \frac{1}{2}\tan^{-1}\left(\frac{t}{4}\right)\right) \, dt\) using a calculator for numerical integration.

Step 6: Interpretation of the Area

The area found represents the difference in distance traveled by the hare and the tortoise between \(t = 0\) and the first intersection time after one hour. A larger area indicates a more significant discrepancy in their speeds over that duration.

Key Concepts

Area Between Curves

Calculating the area between two curves can give you insight into the differences in behavior between two functions over a specific interval. In this exercise, the area between the curves

• represents the difference in distance traveled by the hare and the tortoise over a certain time period.

The two functions, one representing the speed of the hare and the other representing the speed of the tortoise, can have different forms and rates of change.
This integration gives you a quantitative measure of the gap over time. By analyzing this area, you can understand which competitor is leading at different points in time. This concept is very useful when comparing competing factors, like in physics or economics, to determine dominance over an interval.

Intersection Points

Finding intersection points helps determine where two functions have the same output. In our problem, this means identifying when the tortoise and hare travel at equal speeds.
Mathematically, this point is found by setting the two functions equal to one another. This involves solving the equation:

• \( 1 - \cos\left(\frac{\pi t}{2}\right) = \frac{1}{2}\tan^{-1}\left(\frac{t}{4}\right) \)

The solution provides time values where both creatures move at the same speed.
The first intersection after one hour is particularly important in this context as it indicates the next instance where the speeds align following the initial hour. With complex functions, a calculator might be needed to find these points accurately.

Graphing Functions

Graphing functions is a fundamental step in understanding how two functions interact over time. By sketching the graphs of

• \( H(t) = 1 - \cos\left(\frac{\pi t}{2}\right) \)
• \( T(t) = \frac{1}{2}\tan^{-1}\left(\frac{t}{4}\right) \),

you can visualize how the rivelry unfolds over time.
Graphing allows you to see where the two functions intersect and anticipate their behavior beyond hypothetical solutions. The visual representation provides clarity and helps estimate solution intervals, making the identification of intersection points much easier.

Definite Integral

A definite integral helps calculate the total accumulation of a quantity, such as area, between two functions over a certain period. This is crucial in our problem, where we aim to find the area between the speed curves of the tortoise and the hare.
The definite integral:

• \( \int_{0}^{a} \left| H(t) - T(t) \right| \, dt \)

measures the accumulated difference in speed from the start until the first intersection time.
To compute this integral, one typically uses numerical methods or a calculator, especially if the functions involved are not elementary or easy to integrate analytically. This means setting up the integral with the correct limits and evaluating it over the specified interval, gaining insight into the aggregate dominance of one over the other within the given timeframe.