Alright, let's break down the given equation and interpret the coefficient 27.4 step by step.
The given equation is:
[tex]\[ D = 27.4m + 12,100 \][/tex]
Here, [tex]\( D \)[/tex] represents the distance (in millions of miles) of the Voyager 1 spacecraft from the Sun, and [tex]\( m \)[/tex] represents the number of months after January 1, 2015.
### Step-by-Step Solution:
1. Understand the Components:
- The term [tex]\( 27.4m \)[/tex] represents a product of 27.4 and [tex]\( m \)[/tex], where 27.4 is a numerical coefficient.
- The constant term, 12,100, represents the initial distance of Voyager 1 from the Sun on January 1, 2015 (when [tex]\( m = 0 \)[/tex]).
2. Role of Coefficient:
- In any linear equation of the form [tex]\( y = mx + b \)[/tex], the coefficient [tex]\( m \)[/tex] is often referred to as the "slope" or "rate of change." In this specific context, 27.4 is serving as that coefficient.
3. Interpreting the Coefficient 27.4:
- The coefficient 27.4 specifically represents the rate at which the distance [tex]\( D \)[/tex] is increasing as time [tex]\( m \)[/tex] increases.
- Since [tex]\( m \)[/tex] is in months and [tex]\( D \)[/tex] is in millions of miles, 27.4 signifies the change in distance per month.
4. Conclusion:
- The best interpretation of the coefficient 27.4 in this equation is:
[tex]\[
\text{The coefficient 27.4 represents the rate of change of the distance, in millions of miles per month, at which the Voyager 1 spacecraft is moving away from the Sun.}
\][/tex]
Thus, 27.4 million miles per month is how much farther Voyager 1 travels from the Sun for each passing month after January 1, 2015.
