To determine the depth of a toy in a pool depending on the time since it was dropped, we need to establish the constraints and rates that govern its motion. Here, the depth and the rates of falling and sinking are key aspects. Let’s break this scenario down step by step:
1. Pool Depth Constraints:
- The first part of the scenario specifies the possible depths of the pool in meters. The pool can be either 1 meter deep or 2 meters deep.
- Pool depth = 1 meter
- Pool depth = 2 meters
2. Toy Motion Constraints:
- Falling Rate:
- The toy falls at a rate of at least [tex]\(\frac{1}{2}\)[/tex] meter per second. This means the falling rate can be greater than or equal to [tex]\(\frac{1}{2}\)[/tex] meter per second.
- Falling rate [tex]\( \geq 0.5 \)[/tex] meters per second
- Sinking Rate:
- The toy sinks at a rate of no more than [tex]\(\frac{1}{2}\)[/tex] meter per second. This means the sinking rate can be less than or equal to [tex]\(\frac{1}{2}\)[/tex] meter per second.
- Sinking rate [tex]\( \leq 0.5 \)[/tex] meters per second
Given these constraints, the system of inequalities that would represent this scenario includes the following:
1. Depth Constraints:
- [tex]\(d \in \{1, 2\}\)[/tex]
2. Falling Rate Constraint:
- [tex]\( \text{Falling Rate} \geq 0.5\)[/tex]
3. Sinking Rate Constraint:
- [tex]\( \text{Sinking Rate} \leq 0.5\)[/tex]
The constraints that could be part of this scenario are:
- The pool is 1 meter deep.
- The pool is 2 meters deep.
- The toy falls at a rate of at least [tex]\(\frac{1}{2}\)[/tex] meter per second.
- The toy sinks at a rate of no more than [tex]\(\frac{1}{2}\)[/tex] meter per second.
Putting it all together, the constraints identified from the given detailed condition are:
[tex]\[
(\text{Depth of pool} = \text{1 meter}, \text{Depth of pool} = \text{2 meters}, \text{Falling rate} \geq 0.5, \text{Sinking rate} \leq 0.5)
\][/tex]
