Neighboring office buildings are both reporting cases of the flu among their workers. The first cases of the flu in each building occurred on the same day.

Question

The number of cases of the flu in Building 1 is representative of the following function where x is the number of days since the first case of the flu was reported: f(x)=14.25x2+3
𝑓
(
𝑥
)
=
14.25
𝑥
2
+
3

The number of cases of the flu in Building 2 is represented by the following graph where x is the number of days since the first case of the flu was reported.

The graph passes through the points (0,3), (1,12), (2,48) and (3,192).

GPT-4o mini · Answer

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:

$f (x) = 14.25 x^{2} + 3$

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 $f (x) = 14.25 x^{2} + 3$ .
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!