What is the domain of D(t)?
Understand the Problem
The question is asking for the possible values of the function D(t), which represents the distance of the New Horizons spacecraft from Earth in miles after t hours. We need to determine the appropriate domain considering the context of the spacecraft traveling away from Earth.
Answer
The domain of $D(t)$ is all real numbers greater than or equal to 0.
Answer for screen readers
The domain of $D(t)$ is all real numbers greater than or equal to 0.
Steps to Solve
-
Identify the variable t
The variable $t$ represents the time in hours since the New Horizons spacecraft was launched. -
Determine the context of the problem
As the spacecraft travels indefinitely away from Earth at a constant speed of 36,400 miles per hour, we are only interested in the time that is greater than or equal to 0. -
Establish the domain for D(t)
Since the spacecraft cannot be in space for a negative amount of time, the smallest value of $t$ is 0. As time progresses, $t$ can increase indefinitely. -
Formulate the domain
Thus, the appropriate definition of the domain for $D(t)$ includes all real numbers starting from 0 and increasing indefinitely. This can be represented as $[0, \infty)$.
The domain of $D(t)$ is all real numbers greater than or equal to 0.
More Information
In mathematical terms, the domain of a function restricts the possible inputs. Here, since the spacecraft cannot have a negative travel time, the domain is constrained to non-negative values.
Tips
- Assuming that the domain includes negative values: The spacecraft’s journey begins at the launch time ($t=0$), so negative time does not make sense in this context.
- Confusing the distance function with time: Remember to focus on the time variable $t$ when determining the domain.
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