Questions: Lesson 3: Understanding Rational Inputs Cool Down: Flea Treatment Veterinarians use different medications to treat fleas in dogs. Once administered, the medication typically decays exponentially. One treatment decays by half each week. A dog receives 4 ml of medication and the function f gives the number of mL of medicine left after w weeks. 1. Find the amount of medication in the dog's bloodstream 1 week, 2 weeks, and 3 weeks after it is administered. 2. Write an expression to represent the amount of medication in its bloodstream 1 day after it is administered. 3. Explain what f(4/7) means in this context.

Lesson 3: Understanding Rational Inputs Cool Down: Flea Treatment
Veterinarians use different medications to treat fleas in dogs. Once administered, the medication typically decays exponentially. One treatment decays by half each week.
A dog receives 4 ml of medication and the function f gives the number of mL of medicine left after w weeks.
1. Find the amount of medication in the dog's bloodstream 1 week, 2 weeks, and 3 weeks after it is administered.

2. Write an expression to represent the amount of medication in its bloodstream 1 day after it is administered.
3. Explain what f(4/7) means in this context.

Transcript text: Lesson 3: Understanding Rational Inputs Cool Down: Flea Treatment Veterinarians use different medications to treat fleas in dogs. Once administered, the medication typically decays exponentially. One treatment decays by half each week. A dog receives 4 mllof medication and the function $f$ gives the number of mL of medicine left after $w$ weeks. 1. Find the amount of medication in the dog's bloodstream 1 week, 2 weeks, and 3 weeks after it is administered. 2. Write an expression to represent the amount of medication in its bloodstream 1 day after it is administered. 3. Explain what $f\left(\frac{4}{7}\right)$ means in this context.

Solution

Solution Steps

Step 1: Calculate the amount of medication after 1 week, 2 weeks, and 3 weeks

The medication decays by half each week. Starting with 4 mL:

After 1 week: $ 4 \times \frac{1}{2} = 2 $ mL
After 2 weeks: $ 2 \times \frac{1}{2} = 1 $ mL
After 3 weeks: $ 1 \times \frac{1}{2} = 0.5 $ mL

Step 2: Write an expression for the amount of medication after 1 day

Since the medication decays by half each week, the daily decay factor is $ \left(\frac{1}{2}\right)^{\frac{1}{7}} $. Therefore, the amount of medication after 1 day is: \[ 4 \times \left(\frac{1}{2}\right)^{\frac{1}{7}} \]

Step 3: Explain the meaning of $ f\left(\frac{4}{7}\right) $

The function $ f(w) $ represents the amount of medication left after $ w $ weeks. Thus, $ f\left(\frac{4}{7}\right) $ represents the amount of medication left after $ \frac{4}{7} $ weeks, which is approximately 4 days.

Final Answer

Week 1: $ \boxed{2 \text{ mL}} $
Week 2: $ \boxed{1 \text{ mL}} $
Week 3: $ \boxed{0.5 \text{ mL}} $
$ \boxed{4 \times \left(\frac{1}{2}\right)^{\frac{1}{7}}} $
$ \boxed{f\left(\frac{4}{7}\right) \text{ represents the amount of medication left after approximately 4 days.}} $

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