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:

• $t\ge 0$.

Thus, the answer to Part C is $t>-1$ because technically the function is defined for all $t\ge 0$ 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 $t\ge 0$ (or $t>-1$).