Question

f(t)=3(2)^t models the number of people who visited the pumpkin patch since it opened t hours
t f(t)
0
2
5
7
10
Fill out table m, y intercept and domain is t > __

Answer 1

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).

1. **Calculate the values**:

- For \( t = 0 \):
\[
f(0) = 3(2^0) = 3(1) = 3
\]

- For \( t = 2 \):
\[
f(2) = 3(2^2) = 3(4) = 12
\]

- For \( t = 5 \):
\[
f(5) = 3(2^5) = 3(32) = 96
\]

- For \( t = 7 \):
\[
f(7) = 3(2^7) = 3(128) = 384
\]

- For \( t = 10 \):
\[
f(10) = 3(2^{10}) = 3(1024) = 3072
\]

2. **Fill out the table**:
```
t | f(t)
---------------
0 | 3
2 | 12
5 | 96
7 | 384
10 | 3072
```

3. **Y-Intercept**:
The y-intercept occurs at \( t = 0 \). From our calculations:
\[
\text{Y-intercept} = f(0) = 3
\]

4. **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).