Certainly! Let's solve the equation step-by-step to find the amount of aspirin in a patient's body after two hours.
The equation given is:
[tex]\[ a = 500 \cdot \left(\frac{3}{4}\right)^t \][/tex]
Here, [tex]\( a \)[/tex] represents the amount of aspirin left in the body, 500 milligrams is the initial dose of aspirin taken, [tex]\(\frac{3}{4}\)[/tex] is the decay factor representing the proportion of aspirin remaining each hour, and [tex]\( t \)[/tex] is the time in hours.
We need to find the amount of aspirin left after [tex]\( t = 2 \)[/tex] hours. So, we substitute [tex]\( t \)[/tex] with 2 in the equation:
[tex]\[ a = 500 \cdot \left(\frac{3}{4}\right)^2 \][/tex]
First, we calculate the decay factor raised to the power of 2:
[tex]\[ \left(\frac{3}{4}\right)^2 = \frac{3}{4} \cdot \frac{3}{4} = \frac{9}{16} \][/tex]
Now, we multiply this result by the initial amount of aspirin:
[tex]\[ a = 500 \cdot \frac{9}{16} \][/tex]
To perform this multiplication:
[tex]\[ a = 500 \times \frac{9}{16} = 500 \times 0.5625 = 281.25 \][/tex]
Therefore, the amount of aspirin in the patient's body after two hours is 281.25 milligrams.
