The milligrams of aspirin in a person's body is given by the equation 𝑎=500⋅(34)𝑡a=500⋅( 43​ ) t   , where 𝑡t  is the number of hours since the patient took the medicine.2(a)How much aspirin will be in the patient's body after two hours?(b)In the equation, what does the 500 tell us about the situation?Math$$​(c)In the equation, what does the 3443​   tell us about the situation?Math

Sure, let's break down the problem step by step.

Given:

The equation for the milligrams of aspirin in a person's body is: $a = 500 \cdot \left(\frac{3}{4}\right)^t$ where $t$ 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?

To find the amount of aspirin in the body after two hours, we substitute $t = 2$ into the equation:

$a = 500 \cdot \left(\frac{3}{4}\right)^2$

First, calculate $\left(\frac{3}{4}\right)^2$:

$\left(\frac{3}{4}\right)^2 = \frac{3^2}{4^2} = \frac{9}{16}$

Now, multiply this result by 500:

$a = 500 \cdot \frac{9}{16}$

$a = 500 \cdot 0.5625$

$a = 281.25$

So, after two hours, there will be 281.25 milligrams of aspirin in the patient's body.

(b) In the equation, what does the 500 tell us about the situation?

The 500 in the equation represents the initial amount of aspirin (in milligrams) that the patient took. It is the starting value of the aspirin in the body at $t = 0$ hours.

(c) In the equation, what does the $\frac{3}{4}$ tell us about the situation?

The $\frac{3}{4}$ in the equation represents the fraction of aspirin that remains in the body after each hour. It indicates that every hour, the amount of aspirin in the body is reduced to $\frac{3}{4}$ (or 75%) of the amount present at the beginning of that hour. This is a decay factor that shows how the aspirin concentration decreases over time.