Problem 39 - Elementary Algebra

Question

Use numerical evaluation on the equations. Astronomy (Kepler's law of planetary motion) \(P^{2}=k a^{3} . \) Find \(P^{2}\) if \(k=1\) and \(a=4\)

Step-by-Step Solution

Answer

Answer: The value of \(P^{2}\) is 64.

Step 1: Write down the given values

We are given that: - \(k = 1\) - \(a = 4\)

Step 2: Write down Kepler's law of planetary motion equation

Kepler's law of planetary motion equation is given by: \(P^{2} = ka^{3}\).

Step 3: Substitute the given values into the equation

Substitute the given values of \(k\) and \(a\) into the equation: \(P^{2} = (1)(4^{3})\)

Step 4: Calculate \(P^{2}\)

Now, we can calculate \(P^{2}\): \(P^{2} = 1 \cdot 64\) \(P^{2} = 64\) So, the value of \(P^{2}\) is 64.

Key Concepts

Understanding Planetary Motion with Kepler's Law

In the realm of astronomy, Kepler's laws are fundamental in describing planetary motion. Johannes Kepler, a brilliant astronomer, proposed that planets move in elliptical orbits around the sun. His laws help astronomers predict planetary positions and understand their movements without relying on celestial spheres. The law that we're focusing on here is the third law: the square of the orbital period of a planet is proportional to the cube of the semi-major axis of its orbit. This concept can be simplified using the formula \(P^{2} = ka^{3}\). Here, \(P\) is the orbital period, \(k\) is a proportionality constant, and \(a\) is the semi-major axis of the orbit. This equation shows the harmonious and predictive nature of planetary motion, demonstrating a vital relationship between time and space in the cosmos. By using this law, astronomers can calculate how long it takes for a planet to orbit the sun based on its distance.

The Process of Numerical Evaluation

Numerical evaluation is a technique used to find a numerical answer based on a given equation. It involves substituting specific numbers into the variables of an equation and solving it systematically. In the context of Kepler's equation \(P^{2} = ka^{3}\), numerical evaluation allows us to determine the exact value of \(P^{2}\) when given the values of \(k\) and \(a\).Here's the step-by-step approach:

• Identify the known values in the problem; in our case, \(k = 1\) and \(a = 4\).
• Substitute these values into Kepler’s equation: \(P^{2} = (1)(4^{3})\).
• Compute \(4^{3}\) which equals 64, thus \(P^{2} = 1 \cdot 64\).
• The result is \(P^{2} = 64\).

This process of evaluating the expression helps in understanding how variables interact within an equation, reinforcing the importance of accurate substitution and calculation.

Solving Algebraic Equations in Astronomy

Algebraic equations are fundamental for solving various astronomical problems. These equations allow us to quantify relationships between celestial bodies and predict their behavior. In Kepler's third law, the algebraic equation \(P^{2} = ka^{3}\) becomes a powerful tool for astronomers.To solve equations like this, one must:

• Understand the equation structure: identify constants and variables.
• Follow the order of operations carefully: exponents, multiplication, and equality balancing.
• Apply logical reasoning to compute a meaningful result.

This algebraic approach does more than provide numbers. It helps envelop theoretical concepts into practical, calculable results, which is a cornerstone of mathematical science in planetary astronomy. Solving such equations lends insight into how celestial mechanics work, enabling predictions about how planets and other bodies move through space.