The functionf(t) = 3(2)t

Question

models the number of people who visited a pumpkin patch since it opened after t hours.



(1 point)
Part A: Fill out the table below.

t f(t)
0
2
5
7
10
Part B: The y-intercept is
(write your answer as an ordered pair).

Part C: The domain of function given the context of this situation is t >

Answer 1

Let's tackle the problem step by step.

Part A: Fill out the table

The function is given as \( f(t) = 3(2)^t \). We can calculate \( f(t) \) for the specified values of \( t \):

1. When \( t = 0 \): \[ f(0) = 3(2)^0 = 3(1) = 3 \]

2. When \( t = 2 \): \[ f(2) = 3(2)^2 = 3(4) = 12 \]

3. When \( t = 5 \): \[ f(5) = 3(2)^5 = 3(32) = 96 \]

4. When \( t = 7 \): \[ f(7) = 3(2)^7 = 3(128) = 384 \]

5. When \( t = 10 \): \[ f(10) = 3(2)^{10} = 3(1024) = 3072 \]

Now filling out the table:

| t | f(t) | |-----|--------| | 0 | 3 | | 2 | 12 | | 5 | 96 | | 7 | 384 | | 10 | 3072 |

Part B: The y-intercept

The y-intercept occurs when \( t = 0 \). From our calculations:

• The y-intercept is \( (0, f(0)) = (0, 3) \).

Part C: The domain of the function

Given the context that \( t \) represents time in hours since the pumpkin patch opened, the domain consists of non-negative hours. Therefore, we can express the domain as:

• \( t \geq 0 \).

Thus, the answer to Part C is \( t > -1 \) because technically the function is defined for all \( t \geq 0 \) but if we want to indicate that \( t \) can approach 0, we can say \( t > -1 \).

Complete Summary of Answers:

• Part A:

  • Table filled.

• Part B:

  • The y-intercept is \( (0, 3) \).

• Part C:

  • The domain is \( t \geq 0 \) (or \( t > -1 \)).