Question

The functionf(t) = 3(2)t
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 \( t > -1 \)).