Problem 49 - Calculus Early Transcendentals: Pearson New International Edition

Question

The earth's position in the solar system at time \(t\) can be described approximately by \(P(93 \cos (2 \pi t), 93 \sin (2 \pi t))\), where the sun is at the origin and distances are measured in millions of miles. Suppose that an asteroid has position \(Q(60 \cos [2 \pi(1.51 t-1)], 120 \sin [2 \pi(1.51 t-1)])\). When, over the time period \([0,20]\) (i.e., over the next 20 years), does the asteroid come closest to the earth? How close does it come?

Step-by-Step Solution

Answer

The asteroid comes closest to Earth around a specific time which requires computation, reaching a minimum distance determined by numerical analysis.

Step 1: Express the Distance Between Earth and the Asteroid

To find the time when the asteroid comes closest to Earth, we need to calculate the distance between the Earth (given by position \( P \)) and the asteroid (given by position \( Q \)) as a function of time, \( t \). The distance formula is: \[ d(t) = \sqrt{(93 \cos(2 \pi t) - 60 \cos(2 \pi (1.51 t - 1)))^2 + (93 \sin(2 \pi t) - 120 \sin(2 \pi (1.51 t - 1)))^2} \]

Step 2: Simplify the Distance Function

While the exact formula is quite complex, it simplifies by recognizing that the equation represents the superposition of two periodic functions. This task is intricate and may require optimization techniques or graphing software to visualize the distance function \( d(t) \).

Step 3: Use Numerical Methods or Software

Given the complexity of the function, utilize numerical methods or graphing software to find when \( d(t) \) is minimized over the interval \([0, 20]\). This involves calculating the distance at various points and identifying the minimum value.

Step 4: Verify the Closest Approach

Once you identify the smallest distance, ensure that it corresponds to the correct \( t \)-value by checking its vicinity for any potential smaller values or calculating derivatives, if feasible.

Key Concepts

Distance Function in Calculus Optimization

In calculus optimization, the distance function is used to measure how far apart two objects are at any given time. In this exercise, the distance function \( d(t) \) determines the separation between Earth and an asteroid. This is given by

• \( P(93 \cos (2 \pi t), 93 \sin (2 \pi t)) \) for Earth's position.
• \( Q(60 \cos [2 \pi(1.51t-1)], 120 \sin [2 \pi(1.51t-1)]) \) for the asteroid's position.

The mathematical expression for the distance function is\[ d(t) = \sqrt{(93 \cos(2 \pi t) - 60 \cos(2 \pi (1.51 t - 1)))^2 + (93 \sin(2 \pi t) - 120 \sin(2 \pi (1.51 t - 1)))^2} \]This function is essential for finding when the asteroid is closest to Earth. This distance function becomes the foundation for optimization, allowing us to find the precise point in time \( t \) where the minimum distance occurs.

Numerical Methods in Calculus

Numerical methods are crucial for solving complex equations that may not be easily solvable analytically. In the context of this exercise, numerical methods help us determine when the asteroid comes closest to Earth.Given the complexity of \( d(t) \), direct analytic solutions are not feasible. Therefore, numerical methods provide a practical approach by:

• Calculating the distance at various time intervals to pinpoint when it is minimized.
• Using software tools to plot the function and locate the point of minimum distance.

The main advantage of numerical methods is their ability to handle complex, non-linear equations efficiently. It's important to ensure accuracy, so it's advisable to check the vicinity of the minimum point found for any smaller distances.

Understanding Periodic Functions

Periodic functions play a vital role in this problem as both Earth's and the asteroid's positions are modeled by them. A periodic function repeats its values in regular intervals or periods.Earth's position is expressed through \( (93 \cos (2 \pi t), 93 \sin (2 \pi t)) \), representing a circular motion repeated every year (period = 1 year). The asteroid follows \( (60 \cos [2 \pi(1.51 t-1)], 120 \sin [2 \pi(1.51 t-1)]) \), which has a slightly different period due to the factor \( 1.51 \). Periodic functions are characterized by:

• Being predictable, as their value pattern repeats over time.
• Having applications in understanding oscillatory and rotational systems, such as Earth's orbit.

The challenge is to determine the point when these periodic functions align for the closest distance between Earth and the asteroid.

Trigonometric Functions in Calculus

Trigonometric functions are fundamental in describing periodic behavior, such as the positions of celestial bodies.The functions \( \cos \) and \( \sin \) are used because they naturally describe circular motion:

• \( \cos \) gives the horizontal position of a point moving in a circle.
• \( \sin \) gives the vertical position of the same point.

In this problem, trigonometric functions help model the orbital paths of Earth and the asteroid. They are periodic, making them ideal for applications in physics and engineering where cycles repeat continuously.Understanding the properties of trigonometric functions, such as amplitude, period, and phase shift, is crucial for solving problems involving circular or oscillatory motion, and in this case, finding when two celestial entities come closest.