Question

write an equation or function that models the number of bacteria, y, in terms of time, x.
time (secs): 0,1,2,3,4
bacteria: 192,96,48,24,12

Answer 1

The numbers you provided suggest that the bacteria count decreases by half every second. This is indicative of an exponential decay model, which can be expressed in the form:

\[ y = y_0 \cdot a^x \]

where:
- \( y \) is the number of bacteria at time \( x \),
- \( y_0 \) is the initial quantity of bacteria,
- \( a \) is the decay factor,
- \( x \) is the time in seconds.

From your data:
- At \( x = 0 \), \( y = 192 \): This is \( y_0 \).
- The bacteria count halves every second, so \( a = \frac{1}{2} \).

Putting this together, we get:

\[ y = 192 \cdot \left(\frac{1}{2}\right)^x \]

This equation models the number of bacteria \( y \) in terms of time \( x \) in seconds.

To verify:
- When \( x = 0 \): \( y = 192 \cdot \left(\frac{1}{2}\right)^0 = 192 \)
- When \( x = 1 \): \( y = 192 \cdot \left(\frac{1}{2}\right)^1 = 96 \)
- When \( x = 2 \): \( y = 192 \cdot \left(\frac{1}{2}\right)^2 = 48 \)
- When \( x = 3 \): \( y = 192 \cdot \left(\frac{1}{2}\right)^3 = 24 \)
- When \( x = 4 \): \( y = 192 \cdot \left(\frac{1}{2}\right)^4 = 12 \)

The model holds true for all data points.