To analyze the situation, we first need to understand the flu cases reported in each building mathematically.
Building 1
The function for the number of flu cases in Building 1 is given by:
\[ f(x) = 14.25x^2 + 3 \]
where \(x\) is the number of days since the first case of the flu was reported.
Building 2
For Building 2, we have the following points on the graph:
- (0, 3)
- (1, 12)
- (2, 48)
- (3, 192)
These points suggest that the number of flu cases is increasing rapidly. We need to see if we can find a function to represent this growth.
Analyzing Building 2 Data
We can look at the ratios of the y-values to find a pattern:
-
From day 0 to day 1:
- Cases increase from 3 to 12.
- Ratio: \( \frac{12}{3} = 4 \)
-
From day 1 to day 2:
- Cases increase from 12 to 48.
- Ratio: \( \frac{48}{12} = 4 \)
-
From day 2 to day 3:
- Cases increase from 48 to 192.
- Ratio: \( \frac{192}{48} = 4 \)
This suggests that the cases in Building 2 are increasing by a factor of 4 each day.
Finding the Function for Building 2
We notice that the growth appears to be exponential because every day the number of cases multiplies by 4. The exponential function can be expressed as:
\[ g(x) = 3 \times 4^x \]
Let's verify the points:
-
For \(x=0\): \[ g(0) = 3 \times 4^0 = 3 \]
-
For \(x=1\): \[ g(1) = 3 \times 4^1 = 12 \]
-
For \(x=2\): \[ g(2) = 3 \times 4^2 = 48 \]
-
For \(x=3\): \[ g(3) = 3 \times 4^3 = 192 \]
Summary of Functions
- Building 1 has a quadratic growth represented by \( f(x) = 14.25x^2 + 3 \).
- Building 2 has an exponential growth represented by \( g(x) = 3 \times 4^x \).
Conclusions
The two buildings are experiencing different rates of flu case growth, with Building 1 showing a quadratic growth, while Building 2 shows a much faster exponential growth.
If you need further analysis or comparisons between the two buildings (like determining when the number of cases in Building 2 surpasses that in Building 1), we can calculate specific values for any \(x\). Just let me know!
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