To analyze the function \( f(t) = 3(2)^t \) which models the number of people who visited the pumpkin patch since it opened \( t \) hours, let's compute the values for the specific hours (0, 2, 5, 7, and 10).
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Calculate the values:
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For \( t = 0 \): \[ f(0) = 3(2^0) = 3(1) = 3 \]
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For \( t = 2 \): \[ f(2) = 3(2^2) = 3(4) = 12 \]
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For \( t = 5 \): \[ f(5) = 3(2^5) = 3(32) = 96 \]
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For \( t = 7 \): \[ f(7) = 3(2^7) = 3(128) = 384 \]
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For \( t = 10 \): \[ f(10) = 3(2^{10}) = 3(1024) = 3072 \]
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Fill out the table:
t | f(t) --------------- 0 | 3 2 | 12 5 | 96 7 | 384 10 | 3072 -
Y-Intercept: The y-intercept occurs at \( t = 0 \). From our calculations: \[ \text{Y-intercept} = f(0) = 3 \]
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Domain: Since the model represents the number of people visiting a pumpkin patch and time cannot be negative, the domain is: \[ t > 0 \quad (t \geq 0 \text{ for all practical purposes}) \]
Therefore, the filled information is:
- Y-intercept: \( 3 \)
- Domain: \( t \geq 0 \) or \( t > 0 \) depending on context (since we typically think of time as starting from zero).
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