Question

Diego consumes an energy drink that contains caffeine. After consuming the energy drink, the amount of caffeine in Diego's body decreases exponentiall

Exponential growth and decay
ANSWERED
asked 2021-02-25
Diego consumes an energy drink that contains caffeine. After consuming the energy drink, the amount of caffeine in Diego's body decreases exponentially. The 10-hour decay factor for the number of mg of caffeine in Diego's body is 0.2785.
1.What is the 5-hour growth/decay factor for the number of mg of caffeine in Diego's body?
2.What is the 1-hour growth/decay factor for the number of mg of caffeine in Diego's body?
3.If there were 180 mg of caffeine in Diego's body 1.49 hours after consuming the energy drink, how many mg of caffeine is in Diego's body 2.49 hours after consuming the energy drink?

Answers (1)

2021-02-26

Given that:
Diego consumes an energy drink that contains caffeine. The amount of caffeine in Diego's body decreases exponentially.
10-hour decay factor for the number of mg of caffeine in Diego's body is 0.2785
1.
To calculate 1 hours decay factor,
1 hours decay factor \(=(10\text{hours decay factor})^{1/10}\)
\(=(0.2785)^{1/10}\)
\(=0.8799\)
To calculate 5-hour growth/decay factor for the number of mg of caffeine in Diego's body,
5 hours decay factor \(=(1 \text{hours decay factor})^{5}\)
\(=(0.8799)^5\)
\(=0.5274\)
Hence 5-hour \(growth/decay\) factor for the number of mg of caffeine in Diego's body is 0.5274
To calculate 1-hour growth/decay factor for the number of mg of caffeine in Diego's body,
1 hours decay factor \(= 0.8799\) (as calculated above)
Hence 1-hour growth/decay factor for the number of mg of caffeine in Diego's body is 0.8799
3.
Given that:
If there were 180 mg of caffeine in Diego's body 1.49 hours after consuming the energy drink.
The amount of caffeine after 1.49 hours is,
Amount=initialAmount* \((0.8799)^(1.49)\)
180=initialAmount* \((0.8799)^{1.49}\).........(1)
Similarly,
The amount of caffeine after 2.49 hours is,
Amount=initial Amount* \((0.8799)^{2.49}\)
=initial Amount* \((0.8799)^{1.49+1}\)
=initialAmount* \((0.8799)^{1.49} \cdot (0.8799)\)
\(=180 \cdot (0.8799)\)
from equation(1)
\(=158.382\)
Hence 158.382 mg of caffeine is in Diego's body 2.49 hours after consuming the energy drink.

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