Problem 37 - Algebra Form and Function

Question

Find a formula for \(g\) by scaling the output of \(f\). Let \(f(t)\) give the distance in light years to a receding star in year \(t,\) and \(g(t)\) the distance in parsecs. Use the fact that 1 parsec equals 3.262 light years.

Step-by-Step Solution

Answer

Answer: The formula for function g(t) is g(t) = f(t) / 3.262.

Step 1: Write the expression for function \(f(t)\)

Function \(f(t)\) gives the distance in light years to a receding star in year \(t\).

Step 2: Determine the conversion factor from light years to parsecs

According to the given information, 1 parsec (pc) equals 3.262 light years (ly). So, the conversion factor between light years and parsecs can be represented as: 1 pc = 3.262 ly

Step 3: Write down the relationship between the two functions

We want to find a formula for \(g(t)\), which gives the distance in parsecs, based on the given function \(f(t)\), which gives the distance in light years. We can use the conversion factor to create a relationship between the outputs of \(f(t)\) and \(g(t)\): \(g(t) =\) \(\frac{f(t)}{3.262}\)

Step 4: Write the final formula for function \(g(t)\)

The final formula for function \(g(t)\), which gives the distance in parsecs based on the distance in light years given by function \(f(t)\), can be written as: \(g(t) =\) \(\frac{f(t)}{3.262}\) Now, you can use this formula to find the distance in parsecs for any given value of \(t\) by simply scaling the output of the function \(f(t)\).

Key Concepts

Conversion of Units

Converting units is a fundamental skill in math and science, crucial for understanding and communicating measurements accurately. It allows us to express a quantity in different scales, which can be essential for calculations in various contexts.
Imagine you're measuring the distance of a receding star. If the distance is given in light years, but you need it in parsecs, you use the conversion factor to switch between these units.

• The conversion factor acts as a bridge: it's a number that scales one unit to another.
• Knowing that 1 parsec equals 3.262 light years allows us to convert any measurement from light years to parsecs by dividing by this factor.

Remember, the key to successful unit conversion is accuracy. Double-check conversion factors and always ensure your arithmetic is correct.

Function Transformation

Transforming functions involves modifying them to get a new function with desired characteristics. In this context, we scale the output of a function to convert measurements from light years to parsecs.
With function transformation, you manipulate the equations and graphs that represent physical situations. Here, we started with a distance function in light years, denoted as \(f(t)\).

• To transform the function, we always start with an existing function, \(f(t)\), which gives us the starting scenario — distance in light years.
• The transformation is done by applying the conversion factor, leading to a new function, \(g(t)\), that provides distance in parsecs.

The formula \(g(t) = \frac{f(t)}{3.262}\) is an example of a simple linear transformation where we're dividing by a constant to scale the function's output appropriately.

Parsec and Light Year Conversion

Parsecs and light years are units of astronomical distance - both very large, but used in different contexts.

• A light year represents the distance light travels in one year. It is convenient for astronomy because it aligns with the time it takes light to travel vast cosmic spans.
• A parsec is another large unit used primarily by astronomers. It is based on the method of parallax, which involves measuring angles to determine distance.

Understanding their relationship helps in converting between them. We learned that 1 parsec equals 3.262 light years, a crucial factor for any conversion between these measures.
Hence, when you have a distance in light years and need it in parsecs, divide by 3.262. This conversion factor allows astronomers to switch between these scales effortlessly based on their calculation needs.