Answer :
Sure! Let's go through each part of the question step by step.
6. Explicit Rule:
- This is used to find the number of bacteria after [tex]\( n \)[/tex] time increments, where each increment is 30 minutes. Since the bacteria double every 30 minutes, the explicit formula for the number of bacteria after [tex]\( n \)[/tex] increments starting with 150 bacteria is:
[tex]\[
\text{bacteria}(n) = 150 \times 2^n
\][/tex]
This formula tells us exactly how many bacteria there will be after [tex]\( n \)[/tex] half-hour periods.
7. Recursive Rule:
- A recursive rule gives us a way to calculate the number of bacteria at a certain step using the number from the previous step. We start with 150 bacteria, and each half-hour the number doubles, meaning:
[tex]\[
\text{bacteria}(0) = 150
\][/tex]
[tex]\[
\text{bacteria}(n) = 2 \times \text{bacteria}(n-1)
\][/tex]
This shows us how to calculate the number of bacteria at time [tex]\( n \)[/tex] by doubling the amount at time [tex]\( n-1 \)[/tex].
8. Equation for Bacteria After 3 Hours:
- We need to find the number of bacteria after 3 hours. Since each hour has two 30-minute periods, 3 hours would have [tex]\( 3 \times 2 = 6 \)[/tex] increments.
- Using the explicit formula from step 6:
[tex]\[
\text{bacteria}(6) = 150 \times 2^6
\][/tex]
- Given the choices, the correct equation that matches this is:
- Choice B: [tex]\( f(n) = 150 \cdot 2^6 \)[/tex]
Therefore, the explicit rule is [tex]\( \text{bacteria}(n) = 150 \times 2^n \)[/tex], the recursive rule is [tex]\( \text{bacteria}(0) = 150, \text{bacteria}(n) = 2 \times \text{bacteria}(n-1) \)[/tex], and the equation that describes the number of bacteria present 3 hours later is choice B, [tex]\( f(n) = 150 \cdot 2^6 \)[/tex].
6. Explicit Rule:
- This is used to find the number of bacteria after [tex]\( n \)[/tex] time increments, where each increment is 30 minutes. Since the bacteria double every 30 minutes, the explicit formula for the number of bacteria after [tex]\( n \)[/tex] increments starting with 150 bacteria is:
[tex]\[
\text{bacteria}(n) = 150 \times 2^n
\][/tex]
This formula tells us exactly how many bacteria there will be after [tex]\( n \)[/tex] half-hour periods.
7. Recursive Rule:
- A recursive rule gives us a way to calculate the number of bacteria at a certain step using the number from the previous step. We start with 150 bacteria, and each half-hour the number doubles, meaning:
[tex]\[
\text{bacteria}(0) = 150
\][/tex]
[tex]\[
\text{bacteria}(n) = 2 \times \text{bacteria}(n-1)
\][/tex]
This shows us how to calculate the number of bacteria at time [tex]\( n \)[/tex] by doubling the amount at time [tex]\( n-1 \)[/tex].
8. Equation for Bacteria After 3 Hours:
- We need to find the number of bacteria after 3 hours. Since each hour has two 30-minute periods, 3 hours would have [tex]\( 3 \times 2 = 6 \)[/tex] increments.
- Using the explicit formula from step 6:
[tex]\[
\text{bacteria}(6) = 150 \times 2^6
\][/tex]
- Given the choices, the correct equation that matches this is:
- Choice B: [tex]\( f(n) = 150 \cdot 2^6 \)[/tex]
Therefore, the explicit rule is [tex]\( \text{bacteria}(n) = 150 \times 2^n \)[/tex], the recursive rule is [tex]\( \text{bacteria}(0) = 150, \text{bacteria}(n) = 2 \times \text{bacteria}(n-1) \)[/tex], and the equation that describes the number of bacteria present 3 hours later is choice B, [tex]\( f(n) = 150 \cdot 2^6 \)[/tex].
