Question

name date period 6. petri dish a has 20,000 microorganisms and grows by 2,000 microorganisms each week. petri dish b has 1,000 microorganisms and doubles each week. a. which petri dish will have more microorganisms after three weeks? after five weeks? b. here is a graph showing the two petri dishes. which graph corresponds to which situation? explain how you know. c. if you desire a petri dish with more microorganisms, which of the two petri dishes would you choose? explain your reasoning. (from unit 6, lesson 1.) © 2024 illustrative mathematics. includes additional content © by imagine learning

Explanation:

Part (a)

Step 1: Define the growth models

For Petri dish A, the initial number of microorganisms \( P_A(0) = 20000 \) and it grows by 2000 each week. So the formula for the number of microorganisms after \( t \) weeks is \( P_A(t)=20000 + 2000t \).

For Petri dish B, the initial number of microorganisms \( P_B(0)=1000 \) and it doubles each week. So the formula for the number of microorganisms after \( t \) weeks is \( P_B(t)=1000\times2^t \).

Step 2: Calculate for \( t = 3 \) weeks

• For Petri dish A: Substitute \( t = 3 \) into \( P_A(t) \)

\( P_A(3)=20000+2000\times3=20000 + 6000=26000 \)

• For Petri dish B: Substitute \( t = 3 \) into \( P_B(t) \)

\( P_B(3)=1000\times2^3=1000\times8 = 8000 \)
Since \( 26000>8000 \), Petri dish A has more after 3 weeks.

Step 3: Calculate for \( t = 5 \) weeks

• For Petri dish A: Substitute \( t = 5 \) into \( P_A(t) \)

\( P_A(5)=20000+2000\times5=20000 + 10000=30000 \)

• For Petri dish B: Substitute \( t = 5 \) into \( P_B(t) \)

\( P_B(5)=1000\times2^5=1000\times32 = 32000 \)
Since \( 32000>30000 \), Petri dish B has more after 5 weeks.

Part (b)

Petri dish A has a linear growth (since the number of microorganisms increases by a constant amount each week, \( P_A(t)=20000 + 2000t \) is a linear function) and Petri dish B has an exponential growth (since the number of microorganisms multiplies by a constant factor each week, \( P_B(t) = 1000\times2^t\) is an exponential function). In the graph, the points that form a straight line (linear) correspond to Petri dish A, and the points that show a rapid, non - linear increase (exponential) correspond to Petri dish B. We know this because linear functions have a constant rate of change (slope) and exponential functions have a constant multiplicative rate of change.

Part (c)

If we want a Petri dish with more microorganisms, we need to consider the long - term behavior. Petri dish A has a linear growth model \( P_A(t)=20000 + 2000t \), and Petri dish B has an exponential growth model \( P_B(t)=1000\times2^t \). Exponential functions grow faster than linear functions in the long run. As \( t \) becomes large, the term \( 2^t \) in the formula for \( P_B(t) \) will dominate, and \( P_B(t) \) will be greater than \( P_A(t) \). So, if we are considering a time period that is long enough, we should choose Petri dish B. However, if we are only considering a very short time period (shorter than 5 weeks in this case), Petri dish A might have more. But in general, for a desire of more microorganisms (especially in the long - term), Petri dish B is a better choice because of its exponential growth.

Part (a) Answer:

After 3 weeks, Petri dish A (\( P_A(3) = 26000 \)) has more microorganisms than Petri dish B (\( P_B(3)=8000 \)). After 5 weeks, Petri dish B (\( P_B(5) = 32000 \)) has more microorganisms than Petri dish A (\( P_A(5)=30000 \)).

Part (b) Answer:

The linear - looking graph (with a constant slope) corresponds to Petri dish A (since it has linear growth \( P_A(t)=20000 + 2000t \)), and the exponential - looking graph (with increasing slope) corresponds to Petri dish B (since it has exponential growth \( P_B(t)=1000\times2^t \)). We know this because linear functions have a constant rate of change and exponential functions have a constant multiplicative rate of change.

Part (c) Answer:

If we consider the long - term, we should choose Petri dish B. This is because Petri dish B has an exponential growth model (\( P_B(t)=1000\times2^t \)) and exponential functions grow faster than the linear function of Petri dish A (\( P_A(t)=20000 + 2000t \)) in the long run.

Answer:

If we want a Petri dish with more microorganisms, we need to consider the long - term behavior. Petri dish A has a linear growth model \( P_A(t)=20000 + 2000t \), and Petri dish B has an exponential growth model \( P_B(t)=1000\times2^t \). Exponential functions grow faster than linear functions in the long run. As \( t \) becomes large, the term \( 2^t \) in the formula for \( P_B(t) \) will dominate, and \( P_B(t) \) will be greater than \( P_A(t) \). So, if we are considering a time period that is long enough, we should choose Petri dish B. However, if we are only considering a very short time period (shorter than 5 weeks in this case), Petri dish A might have more. But in general, for a desire of more microorganisms (especially in the long - term), Petri dish B is a better choice because of its exponential growth.

Part (a) Answer:

After 3 weeks, Petri dish A (\( P_A(3) = 26000 \)) has more microorganisms than Petri dish B (\( P_B(3)=8000 \)). After 5 weeks, Petri dish B (\( P_B(5) = 32000 \)) has more microorganisms than Petri dish A (\( P_A(5)=30000 \)).

Part (b) Answer:

The linear - looking graph (with a constant slope) corresponds to Petri dish A (since it has linear growth \( P_A(t)=20000 + 2000t \)), and the exponential - looking graph (with increasing slope) corresponds to Petri dish B (since it has exponential growth \( P_B(t)=1000\times2^t \)). We know this because linear functions have a constant rate of change and exponential functions have a constant multiplicative rate of change.

Part (c) Answer:

If we consider the long - term, we should choose Petri dish B. This is because Petri dish B has an exponential growth model (\( P_B(t)=1000\times2^t \)) and exponential functions grow faster than the linear function of Petri dish A (\( P_A(t)=20000 + 2000t \)) in the long run.