A yeast culture grows in a laboratory flask

MATHEMATICS · Multi-part · AP · AP Precalculus · Exponential and Logarithmic Functions · Exponential and Logarithmic Functions

Problem

A yeast culture grows in a laboratory flask. The table shows the measured cell counts

N(t)

at various times

t

(in hours).

$t$ (h)	0	2	4	6	8
$N(t)$ (cells ×10 $^5$ )	5.1	7.0	9.4	12.5	16.3

Summary statistics for the transformed data

\{(t_i,\ln N_i)\}

are provided:

\sum t_i = 20

\sum \ln N_i = 10.5805

\sum t_i\ln N_i = 46.041

\sum t_i^{2}=120

. Assume

N(t)

follows an exponential growth model. (a) Compute the least-squares regression line for

\ln N

versus

t

using the summary statistics. Write your line in the form

\ln N = a+bt

. (b) Express the corresponding exponential growth model for

N(t)

in terms of

t

. (c) Using your model from part (b), estimate the population (in cells ×10

^5

) at

t=10

hours. (d) The observed population at

t=4

hours is

9.4

(cells ×10

^5

). Compute the percentage error between this observation and the value predicted by your model, and comment on what this suggests about the model’s reliability.

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