Task 1
Mo is keeping her potato salad in the fridge until they have dinner. When kept at the refrigerator temperature of 34 ⁰F, a particular strain of Salmonella bacteria has an hourly growth rate of 2 which means that the amount of bacteria or population would double every hour.
Suppose Mo’s potato salad starts in the first hour with one single bacterium. Make a table of values showing the number of bacteria that will be present after each hour for the first six hours using the hourly growth rate of 2.
Write the function that models the data in this table, Bacterial Count as a function of Hours. Remember y = a(b)x where a is the starting value and b is the growth rate.
Merv takes his potato salad with him when he goes camping but doesn’t pack it in a cooler. When he sits down to eat, it has been at 75 ⁰F for 6 hours. At that warmer temperature the Salmonella in the potato salad reproduces much faster, with a growth rate of 8.
Assume Merv’s potato salad also starts in the first hour with one single bacterium. Make a table of values showing the number of bacteria that will be present after each hour for the first six hours using the hourly growth rate of 8.
Write the equation that models the Bacteria Count as a function of Hours with the larger growth rate of 8. Use y = a(b)x where a is the starting value and b is the growth rate.
What is the Bacterial Count for Mo’s potato salad after 6 hours? What is the Bacterial Count for Merv’s potato salad after 6 hours?
Which one, Mo or Merv, is more likely to get sick? Explain your reasoning.
Food rots because different kinds of bacteria eat the food before you do.
To prevent this from happening, you must either eliminate the bacteria or create an environment that inhibits their ability to grow or reproduce.
Radiation, changes in pH, the removal of oxygen, and hot temperatures can be used to kill the bacteria in a food product.
Cold temperatures, dehydration, and chemical preservatives can slow the rate at which bacteria reproduce.
Task 1:
Mo's potato salad:
Hour Bacteria Count
1 1*2^1 = 2
2 1*2^2 = 4
3 1*2^3 = 8
4 1*2^4 = 16
5 1*2^5 = 32
6 1*2^6 = 64
Function for Mo's potato salad: y = 1 * 2^x
Merv's potato salad:
Hour Bacteria Count
1 1*8^1 = 8
2 1*8^2 = 64
3 1*8^3 = 512
4 1*8^4 = 4096
5 1*8^5 = 32768
6 1*8^6 = 262144
Function for Merv's potato salad: y = 1 * 8^x
Bacterial count for Mo's potato salad after 6 hours: 64
Bacterial count for Merv's potato salad after 6 hours: 262144
Merv is more likely to get sick because his potato salad has a significantly higher bacterial count (262144) compared to Mo's potato salad (64). This is due to the warmer temperature of his potato salad, which allows for a faster growth rate of bacteria.
Table for Mo's potato salad:
Hour | Bacterial Count
---------------------
1 | 1
2 | 2
3 | 4
4 | 8
5 | 16
6 | 32
Function for Mo's potato salad: Bacterial Count = 1 * (2^x), where x is the number of hours.
Table for Merv's potato salad:
Hour | Bacterial Count
---------------------
1 | 1
2 | 8
3 | 64
4 | 512
5 | 4096
6 | 32768
Function for Merv's potato salad: Bacterial Count = 1 * (8^x), where x is the number of hours.
The Bacterial Count for Mo's potato salad after 6 hours is 32.
The Bacterial Count for Merv's potato salad after 6 hours is 32768.
Merv is more likely to get sick because his potato salad has a much higher Bacterial Count after 6 hours compared to Mo's potato salad. The higher the Bacterial Count, the greater the risk of getting sick from consuming the contaminated food.
For Mo's potato salad:
Hour 1: 1 bacterium (starting value)
Hour 2: 2 bacteria (doubling the previous hour's count)
Hour 3: 4 bacteria
Hour 4: 8 bacteria
Hour 5: 16 bacteria
Hour 6: 32 bacteria
The function that models the data in this table is: Bacterial Count = 1 * (2^x) where x represents the number of hours.
For Merv's potato salad:
Hour 1: 1 bacterium (starting value)
Hour 2: 8 bacteria (growth rate of 8)
Hour 3: 64 bacteria
Hour 4: 512 bacteria
Hour 5: 4096 bacteria
Hour 6: 32768 bacteria
The function that models the data in this table is: Bacterial Count = 1 * (8^x) where x represents the number of hours.
To find the bacterial count for Mo's potato salad after 6 hours, substitute x = 6 into the function:
Bacterial Count = 1 * (2^6) = 64
To find the bacterial count for Merv's potato salad after 6 hours, substitute x = 6 into the function:
Bacterial Count = 1 * (8^6) = 262,144
Merv's potato salad is more likely to cause sickness as it has a much higher bacterial count after 6 hours compared to Mo's potato salad. The growth rate of 8 bacteria per hour at a higher temperature of 75 ⁰F accelerates bacteria's growth and consequently the risk of getting sick.
To create a table of values showing the number of bacteria for Mo's potato salad, we need to use the equation y = a(b)^x, where y represents the number of bacteria, a represents the starting value (1 bacterium), b represents the growth rate (2), and x represents the number of hours.
Using this equation, we can create a table for the first six hours:
Hour | Bacterial Count
------------------------
1 | 1
2 | 2 * 2 = 4
3 | 4 * 2 = 8
4 | 8 * 2 = 16
5 | 16 * 2 = 32
6 | 32 * 2 = 64
So, the table for Mo's potato salad would look like this:
Hour | Bacterial Count
------------------------
1 | 1
2 | 4
3 | 8
4 | 16
5 | 32
6 | 64
Now, let's create a table for Merv's potato salad. The growth rate is 8, and we'll start with 1 bacterium.
Hour | Bacterial Count
------------------------
1 | 1
2 | 8 * 2 = 16
3 | 16 * 2 = 32
4 | 32 * 2 = 64
5 | 64 * 2 = 128
6 | 128 * 2 = 256
So, the table for Merv's potato salad would look like this:
Hour | Bacterial Count
------------------------
1 | 1
2 | 16
3 | 32
4 | 64
5 | 128
6 | 256
Now, to calculate the bacterial count for Mo's potato salad after 6 hours, we refer to the table and see that it is 64. For Merv's potato salad after 6 hours, the bacterial count is 256.
Based on the bacterial count, Merv is more likely to get sick. This is because the bacteria in Merv's potato salad reproduce at a much faster rate due to the higher growth rate of 8, compared to Mo's potato salad with a growth rate of 2. Higher bacterial counts increase the risk of foodborne illness.