Question 5·Medium·Two-Variable Data: Models and Scatterplots
After an antibiotic is added to a bacterial culture, the number of bacteria decreases by 15% every hour for several hours. What type of model is most appropriate to represent the number of bacteria remaining as a function of time?
For questions about "what type of model" to use, first underline key phrases like "decreases by 15% every hour." Decide if the change is by a fixed amount (suggesting a linear model) or by a fixed percentage or factor (suggesting an exponential model). Then check whether the quantity is going up (growth/positive slope) or down (decay/negative slope) to pick the exact model choice quickly.
Hints
Focus on the key phrase
Look closely at the words "decreases by 15% every hour." What does that tell you about how the amount of bacteria changes from one hour to the next?
Difference vs. percentage
Ask yourself: is the number of bacteria going down by the same number of bacteria each hour, or by the same percentage each hour?
Compare linear and non-linear change
Think about how a linear model behaves: does it use a constant difference or a constant percentage between times? Then ask which type of change is happening in this bacteria situation.
Direction of change
Whatever model type you choose, make sure it matches that the bacteria count is going down over time, not up.
Desmos Guide
Compare linear and percentage-based models
Pick a starting number of bacteria, such as 1000, and enter two expressions in Desmos:
y = 1000 - 150x(this represents losing 150 bacteria every hour, like a linear decrease)y = 1000*(0.85)^x(this represents keeping 85% of the bacteria each hour) Then look at how each graph changes over time.
Check the pattern of change on the graphs
Use a table in Desmos (click on the gear next to each equation and select "table") to see values at . For each model, compare how much drops from one hour to the next, and decide which pattern matches "decreases by 15% every hour" (repeated percentage of the current amount).
Step-by-step Explanation
Identify what is changing and with respect to what
We are modeling the number of bacteria as it changes over time in hours after an antibiotic is added.
So we want a function where the input is time (hours) and the output is the number of bacteria remaining.
Interpret the phrase "decreases by 15% every hour"
"Decreases by 15% every hour" means:
- After 1 hour, the culture has 85% of what it had before (because ).
- After 2 hours, it has 85% of that new amount, and so on.
This is repeated percentage change, so each hour the number is multiplied by (not just subtracting the same fixed number each time).
Connect type of change to type of model
In a linear model, the amount changes by the same fixed number each hour (constant difference), which is not what is described here.
In an exponential model, the amount changes by the same percentage (or factor) each hour (constant ratio), which matches "decreases by 15% every hour".
Decide whether it is growth or decay and choose the model type
Because the number of bacteria is decreasing over time and the change is by a constant percentage, this is an exponential decay situation.
So, the most appropriate model is an exponential decay model.