Question

14
a population of paramecia, p, can be modeled using the exponential function p(t) = 3(2)^t, where t is the number of days since the population was first observed. answer each questions about the population over the course of the first 14 days.
what is the domain of the population in 14 days?
a all real numbers between 0 and 14.
b all real numbers
c all whole numbers between 0 and 14.
d all irrational numbers.

Explanation:

Step1: Understand the context

The problem is about the domain of a function \( P(t) = 3(2)^t \) representing the population of paramecia over \( t \) days (first 14 days). We need to find the domain of \( t \) in this context.

Step2: Analyze the variable \( t \)

• \( t \) represents the number of days since the population was first observed. Over the first 14 days, \( t \) starts at 0 (the start of observation) and goes up to 14 (the end of the first 14 days).
• Also, time in this context (days) can be a real number (e.g., 1.5 days is a valid time for observing population growth, as growth is continuous in the model). So \( t \) can take any real number value from 0 to 14, inclusive.

Step3: Evaluate the options

• Option A: "All real numbers between 0 and 14" matches our analysis, as \( t \) is in \([0, 14]\) (real numbers) for the first 14 days.
• Option B: "All real numbers" is incorrect because \( t \) can't be more than 14 (since we are considering the first 14 days) or negative (time can't be negative in this context).
• Option C: "All whole numbers between 0 and 14" is incorrect because population growth is modeled continuously, so \( t \) can be non - whole numbers (e.g., 0.5 days) as the model is exponential and continuous.
• Option D: "All irrational numbers" is incorrect as \( t \) can be rational (e.g., 1, 2, 3, or 0.25) as well.

Answer:

A. All real numbers between 0 and 14