A biological study tracks two quantities over time (measured in hours) in a…

MATHEMATICS · Multi-part · AP · Calculus AB · Limits and Continuity · Limits and Continuity

Problem

A biological study tracks two quantities over time

t

(measured in hours) in a controlled environment: • The concentration of a nutrient in solution,

N(t)=12e^{0.05t}

milligrams per liter. • The population of a particular microorganism,

P(t)=t^{\,3/2}e^{0.04t}

thousand cells per milliliter. Understanding the long-term relationship between

P

and

N

is essential for predicting when the nutrient supply will become insufficient. (a) Compute

\displaystyle \lim_{t\to\infty}\frac{P(t)}{N(t)}

and interpret what the limit reveals about the balance between population and nutrient concentration as time grows very large. (b) Determine whether the quotient

\dfrac{N(t)}{P(t)}

has a horizontal asymptote. Support your answer with an appropriate limit. (c) Suppose the critical threshold for sustainability is reached when there are 2 thousand microorganisms per milligram of nutrient. Using your work from part (a), explain whether this threshold will be crossed eventually, and if so, describe roughly when in terms of large-

t

behavior rather than an exact numerical time. (d) Scientists propose adding a supplemental nutrient

S(t)=K t^{\,2}e^{0.05t}

milligrams per liter, where

K>0

is a constant to be chosen. Determine all values of

K

for which

\displaystyle \lim_{t\to\infty}\frac{P(t)}{N(t)+S(t)}=0

. Justify your conclusion thoroughly.

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