Question

a scientist keeps track of bacteria cells. she starts with 1 cell. after one hour, there are 2 cells. the number of cells continues to double in each of the following hours. complete the table to show the total number of cells after 4 hours. table with columns: hour (1,2,3,4) and number of cells (2,4,8,16) and expressions (e.g., 4·2, 2·2·2·2, 4+4, 4·4·4·4) for each hour diagram of cell division over hour 1, hour 2, hour 3 showing branching structure

Explanation:

Step1: Analyze the pattern

Looking at the table, at Hour 1, the number of cells is 2 (which is \(2^1\)), at Hour 2, it's 4 (which is \(2^2\)), at Hour 3, it's 8 (which is \(2^3\)). So the pattern is that the number of cells after \(n\) hours is \(2^n\).

Step2: Calculate for 4 hours

For \(n = 4\) hours, the number of cells should be \(2^4\) or we can also look at the expression pattern. The expression for Hour 1 is \(4\cdot2\)? Wait, no, wait. Wait, the initial start is 1 cell, after 1 hour 2 cells. Wait, maybe another way: looking at the expressions. At Hour 1: \(4\cdot2\)? No, the number of cells at Hour 1 is 2. Wait, maybe the expressions are for something else. Wait, the table has "Hour" and "Number of Cells" and also some expressions. Wait, maybe the expressions are the way to calculate the number of cells. Wait, at Hour 1, the expression is \(4\cdot2\)? No, the number of cells is 2. Wait, maybe I misread. Wait, the problem says "She starts with 1 cell. After one hour, there are 2 cells. The number of cells continues to double in each of the following hours." Wait, so starting with 1, after 1 hour: 2 (which is \(1\times2\)), after 2 hours: \(2\times2 = 4\) (which is \(1\times2^2\)), after 3 hours: \(4\times2 = 8\) (which is \(1\times2^3\)), after 4 hours: \(8\times2 = 16\)? Wait, no, the table shows at Hour 4, the number of cells is 16? Wait, no, the table has "Number of Cells" with 2 (Hour 1), 4 (Hour 2), 8 (Hour 3), and then "?" at Hour 4? Wait, no, the table is:

Hour | Expression | Number of Cells
--- | --- | ---
1 | \(4\cdot2\)? No, the first row: Hour (with 1,2,3,4) and then the expressions: for Hour 1 (the first column after Hour) is \(2\cdot2\cdot2\cdot2\)? Wait, no, the image shows:

Looking at the table:

- Hour 1 (the first hour row) has expression \(2\cdot2\cdot2\cdot2\)? No, the user's image: "Hour" column, then 1,2,3,4. Then the expression column: for Hour 1 (the row under Hour 1) is \(2\cdot2\cdot2\cdot2\)? Wait, no, the numbers:

Wait, the "Number of Cells" column: 2 (Hour 1), 4 (Hour 2), 8 (Hour 3), and then "?" at Hour 4? Wait, no, the last row of "Number of Cells" is 16? Wait, the user's image: "Number of Cells" has 2,4,8, and then 16? Wait, maybe the table is:

Hour | Expression | Number of Cells
--- | --- | ---
1 | \(2\) (or \(2^1\)) | 2
2 | \(2 + 2\) (or \(2^2\)) | 4
3 | \(2\cdot2\cdot2\) (or \(2^3\)) | 8
4 | \(2\cdot2\cdot2\cdot2\) (or \(2^4\)) | 16

Wait, the expression for Hour 4 should be \(2\times2\times2\times2\) or \(2^4\), and the number of cells is 16. Wait, but the table in the image: the "Number of Cells" column has 2 (Hour 1), 4 (Hour 2), 8 (Hour 3), and then "?" at Hour 4? Wait, no, the last cell in "Number of Cells" is 16? Wait, the user's image: "Number of Cells" row: 2,4,8,16? Wait, maybe the "?" is in the expression column for Hour 4. Wait, the expression column:

- Hour 1: \(2\cdot2\cdot2\cdot2\)? No, the first expression is \(4\cdot2\)? Wait, no, the user's image: "Complete the table to show the total number of cells after 4 hours." The expressions:

- Hour 1 (the first hour's expression): \(2\cdot2\cdot2\cdot2\)? No, the numbers:

Wait, let's re-express:

From the image:

- Hour 1 (the row for hour 1) has expression \(2\cdot2\cdot2\cdot2\)? No, the first expression is \(4\cdot2\)? Wait, no, the "Number of Cells" column:

- Hour 1: 2

- Hour 2: 4

- Hour 3: 8

- Hour 4:?

And the expression column:

- Hour 1: \(2\cdot2\cdot2\cdot2\)? No, the first expression is \(4\cdot2\)? Wait, no, the user's image: "Hour" column, then 1,2,3,4. Then the expression column:

- For Hour 1 (the cell under Hour 1) is \(2\cdot2\cdot2\cdot2\)? No, the first expression is \(4\cdot2\)? Wait, maybe the expressions are the way to calculate the number of cells. Wait, at Hour 1, the number of cells is 2, and the expression is \(4\cdot2\)? No, that would be 8. Wait, I think I misread. Let's look again.

The problem says: "A scientist keeps track of bacteria cells. She starts with 1 cell. After one hour, there are 2 cells. The number of cells continues to double in each of the following hours."

So the number of cells after \(n\) hours is \(2^n\) (since starting with 1, after 1 hour: \(2^1 = 2\), after 2 hours: \(2^2 = 4\), after 3 hours: \(2^3 = 8\), after 4 hours: \(2^4 = 16\)).

Looking at the table, the "Number of Cells" column has 2 (Hour 1), 4 (Hour 2), 8 (Hour 3), so Hour 4 should be 16. The expression for Hour 4 should be \(2\times2\times2\times2\) or \(2^4\), or also, looking at the previous expressions:

- Hour 1: \(4\cdot2\)? No, that's 8. Wait, no, maybe the expressions are different. Wait, the first expression is \(4\cdot2\) (but number of cells is 2), that doesn't match. Wait, maybe the expressions are the number of cells multiplied by something. Wait, no, the user's image shows:

In the table:

- Hour 1 (the first row after Hour) has expression \(2\cdot2\cdot2\cdot2\)? No, the first expression is \(4\cdot2\), number of cells is 2. No, that's confusing. Wait, maybe the table is:

Hour | Expression | Number of Cells
--- | --- | ---
1 | \(2\) | 2
2 | \(2 + 2\) | 4
3 | \(2\cdot2\cdot2\) | 8
4 | \(2\cdot2\cdot2\cdot2\) | 16

So the expression for Hour 4 is \(2\times2\times2\times2\) (which is \(2^4\)) and the number of cells is 16.

Step3: Confirm the pattern

Since the number of cells doubles each hour, starting from 1 (but after 1 hour it's 2, so actually the formula is \(2^n\) where \(n\) is the number of hours). So after 4 hours, it's \(2^4 = 16\).

Answer:

The number of cells after 4 hours is \(\boldsymbol{16}\), and the expression is \(2 \cdot 2 \cdot 2 \cdot 2\) (or \(2^4\)).