The milligrams of aspirin in a person's body is given by the equation [tex]\( a = 500 \cdot \left(\frac{3}{4}\right)^t \)[/tex], where [tex]\( t \)[/tex] is the number of hours since the patient took the medicine.

a. How much aspirin will be in the patient's body after two hours?



Answer :

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.