The depth of snow after $n$ hours of a snowstorm is represented by the function $f(n+1)=f(n)+0.8$ where $f(0)=2.5$. Which statement describes the sequence of numbers generated by the function?
A. The depth of snow was 0.8 inches when the storm began, and 2.5 inches after the first hour of the storm.
B. The depth of snow was 1.7 inches when the storm began, and 0.8 inches of snow fell each hour.
C. The depth of snow was 2.5 inches when the storm began, and increased by 0.8 inches each hour.
D. The depth of snow was 3.3 inches when the storm began, and 2.5 inches of snow fell in 1 hour.
Answer (2)
The initial depth of snow is f ( 0 ) = 2.5 inches.
The depth increases by 0.8 inches each hour, as described by f ( n + 1 ) = f ( n ) + 0.8 .
Therefore, the depth of snow was 2.5 inches when the storm began, and increased by 0.8 inches each hour.
The correct statement is: The depth of snow was 2.5 inches when the storm began, and increased by 0.8 inches each hour.
Explanation
Understanding the Problem We are given the function f ( n + 1 ) = f ( n ) + 0.8 and the initial condition f ( 0 ) = 2.5 . We need to determine which statement accurately describes the sequence generated by this function.
Finding the Initial Depth The initial condition f ( 0 ) = 2.5 tells us that the depth of snow at the beginning of the storm (when n = 0 ) was 2.5 inches.
Determining the Rate of Increase The function f ( n + 1 ) = f ( n ) + 0.8 tells us that for each subsequent hour ( n + 1 ), the depth of snow increases by 0.8 inches compared to the previous hour ( n ). This means that 0.8 inches of snow falls each hour.
Conclusion Therefore, the depth of snow was 2.5 inches when the storm began, and it increased by 0.8 inches each hour.
Examples
Understanding how snow accumulates can be modeled using mathematical functions like the one in this problem. For example, city planners can use such models to predict how quickly roads will become impassable during a snowstorm, helping them allocate resources for snow removal effectively. Similarly, meteorologists can use more complex models to forecast snowfall rates and total accumulation, aiding in weather warnings and public safety measures. These models help in preparing for and mitigating the impacts of severe weather events.
The depth of snow starts at 2.5 inches when the storm begins and increases by 0.8 inches each hour. Therefore, option C is the correct description of the sequence generated by the function.
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