The function [tex]h(t) = 210 - 15t[/tex] models the altitude of a hot air balloon over time [tex]t[/tex], in minutes.

Explain what [tex]h(10)[/tex] means in the context of the real-world scenario, and how to find its value.



Answer :

Certainly! Let's break this problem down step by step.

### Understanding the Function
The function given is [tex]\( h(t) = 210 - 15t \)[/tex], where:
- [tex]\( h(t) \)[/tex] represents the altitude of the hot air balloon at time [tex]\( t \)[/tex] (in minutes).
- [tex]\( t \)[/tex] is the time in minutes.
- The term [tex]\( 210 \)[/tex] represents the initial altitude of the balloon.
- The term [tex]\( -15t \)[/tex] represents the rate at which the altitude decreases over time.

### Finding [tex]\( h(10) \)[/tex]

Step 1: Interpret the Question

We are asked to find [tex]\( h(10) \)[/tex], which means we need to determine the altitude of the hot air balloon at [tex]\( t = 10 \)[/tex] minutes.

Step 2: Substitute [tex]\( t = 10 \)[/tex] into the Function

Plug [tex]\( t = 10 \)[/tex] into the given function:
[tex]\[ h(10) = 210 - 15 \cdot 10 \][/tex]

Step 3: Perform the Calculation

Let's calculate the value step by step:
1. Multiply [tex]\( 15 \)[/tex] by [tex]\( 10 \)[/tex]:
[tex]\[ 15 \cdot 10 = 150 \][/tex]

2. Subtract this product from [tex]\( 210 \)[/tex]:
[tex]\[ 210 - 150 = 60 \][/tex]

### Conclusion
Therefore, [tex]\( h(10) \)[/tex] equals [tex]\( 60 \)[/tex]. This means that the altitude of the hot air balloon, 10 minutes after it started descending, is [tex]\( 60 \)[/tex] units (the units of altitude are not specified but they might be meters, feet, etc., according to the context of the problem).