Question

Write an equation or function that models the number of bcteria, "y", in terms of time, "x".


Time to Bacteria
0 to 200
1 to 50
2 to 12.5
3 to 3.25

Answer 1

The data form a geometric (exponential) decay: each time unit the bacteria count is multiplied by 1/4. With y(0)=200 the model is

y = 200(1/4)^x.

(Equivalently y = 200 e^{kx} with k = ln(1/4) = −ln 4 ≈ −1.3863, so y = 200 e^{−1.3863 x}.)

Note: this predicts y(3)=200(1/4)^3 = 3.125 (the given 3.25 seems to be a small measurement/rounding difference).