I have a word problem about exponential growth. The problem is...
"At 6am a few bacteria fall into a can of syrup in a broken garbage bag. The conditions of warmth, moisture, and food are perfect for growth and the population doubles every 20 minutes. By 6pm the bacteria are overcrowded and dry and their food is gone."
Q.At what time does the syrup become half full?
Q.A few bacteria search for more food and space. They find three more syrup cans how much of a time reprieve are they given by this find? When will the new cans be depleated?
*I feel like I need the number of the bacteria that fell into the can to be able to answer these questions. Am I over thinking this!?*
fief
To solve these word problems, you do not need the specific number of bacteria that fell into the can initially. Instead, you can solve these problems by understanding the concept of exponential growth and using the given information.
Let's break down each question step by step:
Q1: At what time does the syrup become half full?
To determine when the syrup becomes half full, you need to find the amount of time it takes for the bacteria population to double. Since the population doubles every 20 minutes, it means that after 20 minutes, the population will be twice as large as before.
So, the bacteria population initially starts at 6 am, and it takes 20 minutes for it to double. After another 20 minutes (totaling 40 minutes), it will double again, becoming four times the initial size. Continuing this pattern, after 60 minutes (20 + 20 + 20), it will double again to become eight times the initial size.
By following this pattern, you can determine that after every 20 minutes, the bacteria population doubles. To find when the syrup becomes half full, think of it in terms of the initial size. The population will be half the initial size before the first doubling, which means it will be at 1/2^(1) (1/2 to the power of 1) of the initial size or 1/2.
So, count the number of 20-minute intervals it takes for the population to decrease to one-half of the initial size. In this case, it would require three intervals because 1/2^(3) (1/2 to the power of 3) equals 1/8. Therefore, the syrup becomes half full after 3 intervals of 20 minutes, which is equal to 60 minutes or 6:00 am + 60 minutes = 7:00 am.
Answer: The syrup becomes half full at 7:00 am.
Q2: How much of a time reprieve are they given by finding three more syrup cans? When will the new cans be depleted?
To calculate the time reprieve provided by finding three more syrup cans, you need to determine how the population growth changes with the additional cans.
Initially, with only one can, the population will double every 20 minutes. But with four cans, the bacteria will have four times as much food, leading to a longer timeframe before depletion.
Since the bacteria population doubles every 20 minutes, in the case of four cans, it will take four intervals of 20 minutes for the population to double. So, within the first 20 minutes, the population doubles for the first time, and after another 20 minutes (totaling 40 minutes), it will double a second time, and so on.
To find out how many population doublings occur within a specific timeframe, divide the timeframe by 20. In this case, the timeframe is 12 hours (6 am to 6 pm), which equals 12 x 60 = 720 minutes. So, 720 minutes divided by 20 minutes per doubling gives you 36 doublings within that timeframe.
With each doubling, the population becomes twice as large as before. So, with 36 doublings, the population will be 2^(36) times the initial size.
Answering the second part of the question, to determine when the new cans will be depleted, you need to find how many doublings it would take for the population to exceed the available resources. In this case, the available resources are three additional cans of syrup.
So, the population can double three times (to become eight times the initial size) before the cans are depleted, as there are three cans available. After three doublings, the population size is 2^(3) times the initial size.
To find out when the new cans will be depleted, you need to determine the time it takes for three doublings to occur. Each doubling takes 20 minutes, so three doublings will take 3 x 20 = 60 minutes.
Answer: The new cans will be depleted after 60 minutes or 6:00 am + 60 minutes = 7:00 am.
Therefore, the bacteria are given a time reprieve of 60 minutes by finding three more syrup cans.