Mika is doing an experiment with bacteria .She finds that , provided there is enough space and food , the population doubles every 2 hours. The population in hour y+10 will be how many times the population in hour y ?

Options

A. 5
B. 10
C. 16
D. 32

Answer

To solve the problem step by step, we will analyze the growth of the bacteria population over time and find the ratio of the population at hour \( y + 10 \) to the population at hour \( y \).

### Step 1: Understand the population growth
The population doubles every 2 hours. This means that if we start with an initial population \( P_0 \), after 2 hours the population will be \( 2P_0 \), after 4 hours it will be \( 4P_0 \), and so on.

### Step 2: Define the initial population
Let's assume the initial population at hour 0 is \( P_0 = x \).

### Step 3: Calculate the population at hour \( y + 10 \)
To find the population at hour \( y + 10 \), we need to determine how many 2-hour intervals fit into \( y + 10 \):
- The time from hour 0 to hour \( y + 10 \) is \( y + 10 \) hours.
- The number of 2-hour intervals in \( y + 10 \) hours is \( \frac{y + 10}{2} \).

Thus, the population at hour \( y + 10 \) is:
\[ P(y + 10) = x \cdot 2^{\frac{y + 10}{2}} \]

### Step 4: Calculate the population at hour \( y \)
Similarly, for hour \( y \):
- The time from hour 0 to hour \( y \) is \( y \) hours.
- The number of 2-hour intervals in \( y \) hours is \( \frac{y}{2} \).

Thus, the population at hour \( y \) is:
\[ P(y) = x \cdot 2^{\frac{y}{2}} \]

### Step 5: Find the ratio of populations
Now, we need to find how many times the population at hour \( y + 10 \) is compared to the population at hour \( y \):
\[ \text{Ratio} = \frac{P(y + 10)}{P(y)} = \frac{x \cdot 2^{\frac{y + 10}{2}}}{x \cdot 2^{\frac{y}{2}}} \]

### Step 6: Simplify the ratio
The \( x \) cancels out:
\[ \text{Ratio} = \frac{2^{\frac{y + 10}{2}}}{2^{\frac{y}{2}}} = 2^{\frac{y + 10}{2} - \frac{y}{2}} = 2^{\frac{10}{2}} = 2^5 = 32 \]

### Final Answer
Thus, the population at hour \( y + 10 \) is **32 times** the population at hour \( y \).

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