1. **What does \( t \) represent?**
In the function \( b(t) = -0.32t^2 + 2.7t + 253 \), \( t \) represents the number of years since 2008. So \( t = 0 \) corresponds to the year 2008, \( t = 1 \) to the year 2009, and so on.
2. **What is the domain for this function?**
The domain of this function would be all non-negative integers starting from 0 because those represent the number of years since 2008. Hence, \( t \) can be \( 0, 1, 2, 3, \ldots \). Realistically, \( t \) cannot be negative because you cannot have a negative count of years since 2008.
3. **Which range values would not make sense for this function?**
Since the function represents the population of bobcats, only non-negative values of \( b(t) \) make sense. If \( b(t) \) were negative, it would not be realistic since you cannot have a negative population.
To determine the values of \( t \) where \( b(t) \) is non-negative, we solve for \( t \) when \( b(t) = 0 \):
\[
-0.32t^2 + 2.7t + 253 = 0
\]
Using the quadratic formula \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = -0.32 \), \( b = 2.7 \), and \( c = 253 \):
\[
t = \frac{-2.7 \pm \sqrt{(2.7)^2 - 4(-0.32)(253)}}{2(-0.32)}
\]
\[
t = \frac{-2.7 \pm \sqrt{7.29 + 323.84}}{-0.64}
\]
\[
t = \frac{-2.7 \pm \sqrt{331.13}}{-0.64}
\]
\[
t = \frac{-2.7 \pm 18.2}{-0.64}
\]
Solving for both roots yields approximately:
\[
t \approx \frac{-2.7 + 18.2}{-0.64} \approx -24.22
\]
\[
t \approx \frac{-2.7 - 18.2}{-0.64} \approx 32.81
\]
The realistic value of \( t \) would be between 0 and 32.81 for non-negative population. Thus, realistic values for \( t \) are in the range \( [0, 32] \) (since \( t \) is integer).
4. **Would the graph be continuous or discrete, and why?**
The graph would be continuous because the mathematical function \( b(t) \) defined by a quadratic equation is continuous. However, in practical terms, the population count at any given year is a discrete number, typically represented by whole numbers. Therefore, while the mathematical model is continuous, the actual data points representing population for each year would be discrete.
The population of bobcats in northern Arizona since 2008 can be modeled using the function b(t) = –0.32t2 + 2.7t + 253.
What does t represent?
What is the domain for this function?
Which range values would not make sense for this function?
Would the graph be continuous or discrete, and why?
1 answer