Certainly! Let's delve into the problem step by step.
1. Understanding the Variables:
- [tex]\( t \)[/tex] represents the time in minutes. It quantifies how long it has been since we started observing the balloon’s descent.
- [tex]\( a(t) \)[/tex] represents the altitude of the balloon in meters at a specific time [tex]\( t \)[/tex]. This tells us how high the balloon is above the ground at that moment.
2. Function Analysis:
The function given is [tex]\( a(t) = 210 - 15t \)[/tex]. This linear function has two components:
- The constant term [tex]\( 210 \)[/tex] represents the initial altitude of the balloon before it begins descending.
- The term [tex]\( -15t \)[/tex] represents the rate at which the balloon’s altitude decreases over time. Specifically, the balloon is descending at a rate of 15 meters per minute.
3. Calculating [tex]\( a(5.5) \)[/tex]:
- To find the altitude of the balloon at [tex]\( t = 5.5 \)[/tex] minutes, we substitute [tex]\( t = 5.5 \)[/tex] into the function.
[tex]\[ a(5.5) = 210 - 15 \times 5.5 \][/tex]
- Substituting the time ([tex]\( t \)[/tex]) into the equation:
[tex]\[ a(5.5) = 210 - 15 \times 5.5 \][/tex]
[tex]\[ a(5.5) = 210 - 82.5 \][/tex]
- Simplifying the equation:
[tex]\[ a(5.5) = 127.5 \][/tex]
So, when [tex]\( t = 5.5 \)[/tex] minutes, the altitude of the balloon [tex]\( a(t) \)[/tex] is 127.5 meters.
### Summary:
- [tex]\( t \)[/tex] represents the time in minutes.
- [tex]\( a(t) \)[/tex] represents the altitude of the balloon in meters at time [tex]\( t \)[/tex].
- [tex]\( a(5.5) \)[/tex] gives the altitude of the balloon after 5.5 minutes, which is 127.5 meters.
This means that 5.5 minutes into the observation, the balloon is at an altitude of 127.5 meters above the ground.
