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.