Question
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When the county fair opened its gates, 68 people entered the fairgrounds. After one hour, there were 1.5 times as many people on the fairgrounds as when the gates opened. After two hours, there were 1.5 times as many people on the fairgrounds as the previous hour. If this pattern continues, write the function representing the number of people, f(x), at the fair x hours after the gates open.
When the county fair opened its gates, 68 people entered the fairgrounds. After one hour, there were 1.5 times as many people on the fairgrounds as when the gates opened. After two hours, there were 1.5 times as many people on the fairgrounds as the previous hour. If this pattern continues, write the function representing the number of people, f(x), at the fair x hours after the gates open.
Answers
GPT-4o mini
To determine the number of people at the fair x hours after the gates open, we start with the information provided.
1. When the gates opened (x = 0), there were 68 people:
\( f(0) = 68 \)
2. After one hour, there are 1.5 times as many people as when the gates opened:
\( f(1) = 1.5 \times f(0) = 1.5 \times 68 = 102 \)
3. After two hours, there are again 1.5 times as many people as the previous hour:
\( f(2) = 1.5 \times f(1) = 1.5 \times 102 = 153 \)
We can observe that the number of people increases by a factor of 1.5 each hour. Thus, we can express the function in terms of \( x \):
- For 0 hours: \( f(0) = 68 \)
- For 1 hour: \( f(1) = 68 \times (1.5)^1 \)
- For 2 hours: \( f(2) = 68 \times (1.5)^2 \)
Hence, we see a consistent pattern, and we can generalize this to:
\[
f(x) = 68 \times (1.5)^x
\]
This function represents the number of people at the fair x hours after the gates open. Thus, the function is:
**f(x) = 68 * (1.5)^x**
1. When the gates opened (x = 0), there were 68 people:
\( f(0) = 68 \)
2. After one hour, there are 1.5 times as many people as when the gates opened:
\( f(1) = 1.5 \times f(0) = 1.5 \times 68 = 102 \)
3. After two hours, there are again 1.5 times as many people as the previous hour:
\( f(2) = 1.5 \times f(1) = 1.5 \times 102 = 153 \)
We can observe that the number of people increases by a factor of 1.5 each hour. Thus, we can express the function in terms of \( x \):
- For 0 hours: \( f(0) = 68 \)
- For 1 hour: \( f(1) = 68 \times (1.5)^1 \)
- For 2 hours: \( f(2) = 68 \times (1.5)^2 \)
Hence, we see a consistent pattern, and we can generalize this to:
\[
f(x) = 68 \times (1.5)^x
\]
This function represents the number of people at the fair x hours after the gates open. Thus, the function is:
**f(x) = 68 * (1.5)^x**