The data in the table can best be described as exponential decay because there is a consistent percentage decrease over time.
When modeling the volume of the water storage tank over the weeks, this type of decreasing pattern is typically represented by an exponential decay function. The general form of an exponential decay function is:
[tex]\[ V(t) = V_0 \cdot e^{-kt} \][/tex]
where:
- [tex]\( V(t) \)[/tex] is the volume at time [tex]\( t \)[/tex],
- [tex]\( V_0 \)[/tex] is the initial volume,
- [tex]\( k \)[/tex] is the decay constant,
- [tex]\( t \)[/tex] is the time.
From analyzing the data, we find that an exponential decay model fits the data points. By fitting the model to the given data, we can determine the parameters [tex]\( V_0 \)[/tex] and [tex]\( k \)[/tex].
Given the data:
- At [tex]\( t = 1 \)[/tex] week, [tex]\( V(1) = 12.0 \)[/tex] cubic meters,
- At [tex]\( t = 2 \)[/tex] weeks, [tex]\( V(2) = 6.1 \)[/tex] cubic meters,
- At [tex]\( t = 3 \)[/tex] weeks, [tex]\( V(3) = 3.0 \)[/tex] cubic meters,
- At [tex]\( t = 4 \)[/tex] weeks, [tex]\( V(4) = 1.6 \)[/tex] cubic meters,
- At [tex]\( t = 5 \)[/tex] weeks, [tex]\( V(5) = 0.8 \)[/tex] cubic meters.
By fitting the exponential decay model to these data points, we get:
- The initial volume [tex]\( V_0 \approx 23.731 \)[/tex] cubic meters, and
- The decay constant [tex]\( k \approx 0.682 \)[/tex].
Therefore, the model describing the volume of the water storage tank over time is:
[tex]\[ V(t) = 23.731 \cdot e^{-0.682t} \][/tex]
This verifies that the volume decreases in an exponential manner, confirming that the data in the table can best be described as exponential decay.
