A research laboratory releases a nutrient into a closed culture every hour

MATHEMATICS · Multi-part · A Level · Mathematics · Sequences and series · Sequences and series

Problem

A research laboratory releases a nutrient into a closed culture every hour. The amount released at hour

n

is modeled by

a r^{n}

grams, where

n=0,1,2,\dots

. After all future hours have elapsed, the total amount of nutrient released must equal

900

grams. (a) Express the total amount of nutrient released as an infinite series and identify the parameter(s) on which that total depends. (b) State the condition that the ratio

r

must satisfy for the infinite series you wrote in part (a) to converge, and justify your answer. (c) Derive an equation that relates

a

and

r

to the required total of

900

grams, then solve this equation for

a

in terms of

r

. (d) If the laboratory stipulates that

r=0.60

, calculate the corresponding value of

a

and interpret its meaning in the context of the nutrient release schedule.

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