In this problem, we are dealing with the function [tex]\( a(t) = 210 - 15t \)[/tex], which describes the altitude of a hot air balloon as it approaches the ground, where [tex]\( t \)[/tex] is the time in minutes.
Let's address each part of the question in detail.
1. What does [tex]\( t \)[/tex] represent?
- The variable [tex]\( t \)[/tex] represents the time in minutes. It's the time elapsed since a specific starting point, such as when the balloon began its descent.
2. What does [tex]\( a(t) \)[/tex] represent?
- The function [tex]\( a(t) \)[/tex] represents the altitude of the balloon as it approaches the ground, measured in meters. Given [tex]\( t \)[/tex] minutes, [tex]\( a(t) \)[/tex] provides the height of the balloon above the ground at that particular time.
3. What information will [tex]\( a(5.5) \)[/tex] give?
- To determine the information provided by [tex]\( a(5.5) \)[/tex], we will substitute [tex]\( t = 5.5 \)[/tex] into the function [tex]\( a(t) \)[/tex]. This will give us the altitude of the balloon 5.5 minutes after the descent began.
Substituting [tex]\( t = 5.5 \)[/tex] into the function [tex]\( a(t) \)[/tex]:
[tex]\[ a(5.5) = 210 - 15 \cdot 5.5 \][/tex]
Calculating the value:
[tex]\[ a(5.5) = 210 - 82.5 \][/tex]
[tex]\[ a(5.5) = 127.5 \][/tex]
So, [tex]\( a(5.5) \)[/tex] provides the altitude of the balloon after 5.5 minutes have passed. The altitude of the balloon at 5.5 minutes is 127.5 meters.
