What is the range of S(n)?
Understand the Problem
The question is asking for the range of the function S(n), which represents the amount of money made from selling n passes, given that each pass sells for $105 and a maximum of 500 passes can be sold this season. The focus is on identifying the correct set of values that correspond to this range based on the number of passes sold.
Answer
The range of $S(n)$ is all multiples of $105$ from $0$ to $52,500$.
Answer for screen readers
The range of $S(n)$ is all multiples of $105$ from $0$ to $52,500$.
Steps to Solve
- Understand the function S(n)
The function $S(n)$ represents the total money made from selling $n$ passes, where each pass sells for $105$. Thus, we can express $S(n)$ as: $$ S(n) = 105n $$
- Determine the range of n
Since the maximum number of passes sold is 500, $n$ can take any whole number value from 0 to 500. Therefore, we have: $$ n = 0, 1, 2, \ldots, 500 $$
- Calculate the minimum and maximum values of S(n)
Using the expression for $S(n)$, we can calculate:
- Minimum value (when $n = 0$): $$ S(0) = 105 \times 0 = 0 $$
- Maximum value (when $n = 500$): $$ S(500) = 105 \times 500 = 52500 $$
- Identify the set of values in the range
The function $S(n)$ can produce values from $0$ to $52500$ in increments of $105$. Thus, the range of $S(n)$ consists of: $$ S(n) = 0, 105, 210, \ldots, 52500 $$
These values are all multiples of $105$ within the specified range.
- Specify the correct answer
The range of $S(n)$ can be summarized as the set of all multiples of $105$ from $0$ to $52500$.
The range of $S(n)$ is all multiples of $105$ from $0$ to $52,500$.
More Information
The maximum revenue possible from selling all 500 passes is $52,500$, achieved when every pass is sold at $105$. The multiples of $105$ represent all possible total sales amounts based on the number of passes sold.
Tips
- Confusing the maximum revenue with the total number of passes instead of understanding that $S(n)$ depends on $n$ times the price per pass.
- Overlooking the fact that total sales can start from zero, leading to a misinterpretation of the valid range of the function.
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