Algebra
- The difference between exponential and linear functions is in the rate at which the change occurs. Therefore, the scenario in plan A can be modelled with a linear function because the rate of infection occurs at a constant rate of 14 people every day. Conversely, the scenario in plan B can be modelled with an exponential function because the rate of change in infection doubles at consequitive intervals.
- The scenario in plan A can be modelled by function 2, P(t) =14t +4. Function 2 is right since it models a linear relationship. In this scenario, the change in the number of infected people has a linear relationship with the number of days. Therefore, the slope of the graph (14) represents the rate of infection. The y-intercept (4) represents the number of people with the virus at the beginning of the process.
The scenario in Plan B can be modelled by function 1 [P(t) =4(2)t ]. An exponential function models a situation where a small change in the independent variable causes a greater change in the dependent variable (Ciarlet 31). The function effectively represents this scenario because the number of infected persons doubles with each passing day. This can be proven by inputting different values of t in the exponential function.
- For any of the scenarios, the expression implies that the number of people with the virus on day four will be 60.
- To determine how many people will have the virus each day, input the values of “t” (the number of the days) in the equations P(t) =14t +4 and P(t) =4(2)t for plan A and plan B, respectively.
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| Day | Population with the virus plan A | Population with the virus plan B |
| 0 | 4 | 4 |
| 1 | 18 | 8 |
| 2 | 32 | 16 |
| 3 | 46 | 32 |
| 4 | 60 | 64 |
| 5 | 74 | 128 |
| 6 | 88 | 256 |
| 7 | 102 | 512 |
| 8 | 116 | 1024 |
Table 1: Number of peoples with the virus each day
- Graphs of the population with the virus over the first six days for plan A and plan B.
Figure 1: Population with the virus in plan A
In Figure 1, the data lies on the straight line. Therefore, the rate of change remains constant.
Figure 2: Population with the virus in plan B
Figure 2 shows that the population with the virus in plan B increases exponentially.
- To find the average rate of change, the change in the output value is divided by the change in the input value.
The average rate of change =
Plan A: From the table, in day 0 the population with the virus was 4. On day 3 the population was 46. The output (population with the virus) changes by 42 people. Simultaneously, the input (days) changes by 3 days. Therefore, the average rate of change from day 0 to day 3 is calculated as shown below.
= 14 people per day
Plan B: From the table, in day 0 the population with the virus was 4. On day 3 the population was 32. Therefore, the average rate of change in plan B is calculated as shown below.
people per day
The average rate of change helps to find the magnitude of change in the y-value (new infections) in relation to the changes in the x-values (number of days) (Lo and Kratky 51). In both plans, the rate of change is positive implying that the infection spreads in both scenarios. However, the fact that the rate of change in plan B is lower than in plan A shows that plan B results in a lower spread of the virus over the first 3 days.
- Plan A: On day 0, the population with the virus was 4. On day 8, the infected population was 116.
The average rate of change =
The average rate of change = = 14 people per day.
Plan B: On day 0, the population with the virus was 4. On day 8 the population was 1024.
The average rate of change = = 127.5 people per day.
The average rate of change shows the rate of spread of the virus. Since the average rate of change in plan A was less than the average rate in plan B, the spread of the virus in plan A is lower than in plan A over the first 8 days.
- h.
In both scenarios, the rate of infection increases. The scenario in plan A indicates that the infections will increase at a constant rate of 14 people per day. The scenario can be modeled by a linear function. The model implies that the virus infection and the number of days have a linear relationship. On the other hand, the scenario in plan B indicates that the number of infections doubles with each passing day. The scenario can be modeled by an exponential function.
Both linear and exponential functions indicate the rate of change between different intervals. The rate of change helps to determine the best plan to implement. Over the first 3 days, the rate of change in plan B is lower than in plan A. This shows that plan B results in lower virus infections in the short run. However, the best plan is the one that demonstrates the least rate of virus spread in the long run. Since the average rate of change in plan A was less than the average rate in plan B over the first 8 days, plan A results in a lower virus spread in the long run. Therefore, plan A is the best plan to implement.
Works Cited
Ciarlet, Philippe G. Linear and nonlinear functional analysis with applications. Vol. 130. Siam, 2013.
Lo, Jane-Jane, and James L. Kratky. “Looking for connections between linear and exponential functions.” MatheMatics teacher 106.4 (2012): 295-301.