Sure, let's walk through the process of understanding and finding the value of [tex]\( h(10) \)[/tex] within the context of the given real-world scenario.
### Understanding the Function
The function given is:
[tex]\[ h(t) = 210 - 15t \][/tex]
This function models the altitude (in meters) of a hot air balloon over time [tex]\( t \)[/tex] (in minutes). Specifically, [tex]\( h(t) \)[/tex] represents the altitude of the balloon at a given time [tex]\( t \)[/tex].
### Breaking Down the Function
- The constant term [tex]\( 210 \)[/tex] represents the initial altitude of the hot air balloon when [tex]\( t = 0 \)[/tex]. This means that at time zero (the starting point), the balloon is at 210 meters.
- The term [tex]\( -15t \)[/tex] indicates that the altitude of the balloon decreases by 15 meters for each minute that passes. So, for every minute [tex]\( t \)[/tex], we subtract [tex]\( 15t \)[/tex] meters from the initial altitude.
### Evaluating [tex]\( h(10) \)[/tex]
Now, we want to find the altitude of the balloon at [tex]\( t = 10 \)[/tex] minutes. To do this, we substitute [tex]\( t \)[/tex] with 10 in the function:
[tex]\[ h(10) = 210 - 15 \cdot 10 \][/tex]
We perform the multiplication and subtraction operations step-by-step:
1. Multiply 15 by 10:
[tex]\[ 15 \cdot 10 = 150 \][/tex]
2. Subtract the result from 210:
[tex]\[ 210 - 150 = 60 \][/tex]
### Conclusion
So, [tex]\( h(10) \)[/tex] represents the altitude of the hot air balloon at 10 minutes. After calculating, we find that:
[tex]\[ h(10) = 60 \][/tex]
This means that, at 10 minutes, the altitude of the hot air balloon is 60 meters.
