A scientist is studying the decay of a certain substance after being exposed to two different treatments. Under treatment A, the
substance continuously decays at a rate of 4% daily. Under treatment B, another sample of the same substance continuously decays at a rate of 6.2% daily.A second scientist comes to record the amount remaining each day and only knows that there was initially less than 300 grams of the substance undergoing treatment A and at most 400 grams of the substance undergoing treatment B.What system of inequalities can be used to determine t, the number of days after which the remaining amount of each sample,y, in grams, is the same?
2 answers:
Answer:
- 0 < y < 300·0.96^t
- 0 < y ≤ 400·0.938^t
Step-by-step explanation:
When a quantity changes exponentially by a fraction r in some time period t, the quantity is multiplied by 1+r in each period. That is the quantity (y) as a function of t can be described by ...
y = y0·(1+r)^t
where y0 is the initial quantity (at t=0).
Here, the problem statement gives us two quantities and their respective rates of change.
<u>Treatment A</u>
y0 < 300, r = -0.04, so the remaining amount is described by ...
y < 300·0.96^t
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Treatment B
y0 ≤ 400, r = -0.062, so the remaining amount is described by ...
y ≤ 400·0.938^t
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When we graph these, we realize these inequalities allow the quantity of each substance to be less than zero. Mathematically, those quantities will approach zero, but not equal zero, so we can put 0 as a lower bound on the value of y in each case:
- 0 < y < 300·0.96^t
- 0 < y ≤ 400·0.938^t
_____
<em>Comment on these inequalities</em>
We suspect your answer choices will not be concerned with the lower bound on y.
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