Biologists model the mass (in grams) of a bacterial culture over time using

MATHEMATICS · Multi-part · A Level · Mathematics · Exponentials and logarithms · Exponentials and logarithms

Problem

M(t)=A e^{-k t}

, where

A

is the initial mass and

k>0

is an unknown decay rate. The culture is measured every hour for six hours; the data are recorded below.

Time $t$ (h)	0	1	2	3	4	5	6
Measured mass $M_{\text{obs}}$ (g)	5.00	3.95	3.10	2.46	1.94	1.56	1.25

Because the initial reading is precise, set

A=5.00\text{ g}

. You must determine the decay rate

k

that best matches the remaining observations. (a) Write an explicit expression

S(k)

for the total squared deviation between the model

M(t)=5.00 e^{-k t}

and the observed masses for

t=1

through

t=6

. (b) Derive

S'(k)

and set it equal to zero. Show all calculus steps leading to an equation that the optimal

k

satisfies. (c) Using technology or justified numerical estimation, solve the equation from part (b) to three decimal places. State the value of

k

with appropriate units. (d) Based on your value of

k

, predict the culture’s mass at

t=8\text{ h}

. Comment on the reliability of this prediction, considering the spacing of the data and the exponential model.