Let's break down the problem step by step to determine the correct formula for Lexi's medication in her system.
We know the following:
1. Sofia takes 10 mg of a medicine whose concentration decreases by a factor of one-half every day. The amount [tex]\( S(t) \)[/tex] of medication in Sofia's system after [tex]\( t \)[/tex] days is given by:
[tex]\[ S(t) = 10 \left(\frac{1}{2}\right)^t \][/tex]
2. Lexi takes the same medicine, but she starts taking it 4 days after Sofia. We need to find the formula for the amount of medication [tex]\( L(t) \)[/tex] in Lexi's system on day [tex]\( t \)[/tex].
Here’s the step-by-step solution:
### Step 1: Understand the Day Counting
- Let's denote [tex]\( t \)[/tex] as the number of days since Sofia took the medicine.
- Since Lexi starts 4 days later, on Sofia's day [tex]\( t \)[/tex], for Lexi it is [tex]\( t - 4 \)[/tex] days since she took the medicine.
### Step 2: Apply the Decrease Factor to Lexi's Medication
- Since the concentration decreases by a factor of one-half every day, the amount of medication in Lexi's system [tex]\( L \)[/tex] can be modeled similarly to Sofia’s, but we need to account for the 4-day delay.
### Step 3: Write the Formula for Lexi
- On day [tex]\( t \)[/tex] for Sofia, it is [tex]\( t - 4 \)[/tex] days since Lexi took the medicine. Therefore, the amount of medication in Lexi’s system is given by:
[tex]\[ L(t) = 10 \left(\frac{1}{2}\right)^{t - 4} \][/tex]
### Step 4: Compare with Given Choices
Let’s compare this result with the given multiple-choice options to determine which matches our formula:
1. [tex]\( L = 10 \left(\frac{1}{2}\right)^t \)[/tex] – This is for someone who took the medicine on day zero, same as Sofia.
2. [tex]\( L = 10 \left(\frac{1}{2}\right)^{t + 4} \)[/tex] – This would be for someone who took the medicine 4 days before Sofia, which is not the case.
3. [tex]\( L = 10 \left(\frac{1}{2}\right)^{t - 4} \)[/tex] – This matches our derived formula.
4. [tex]\( L = 10 \left(\frac{1}{2}\right)^{4t} \)[/tex] – This doesn't correlate to any logical delay or decrease model for Lexi.
Thus, the correct choice is:
[tex]\[ \boxed{L = 10 \left(\frac{1}{2}\right)^{t - 4}} \][/tex]
