The function 

Upper N left parenthesis t right parenthesis equals StartFraction 0.7 t plus 800 Over 7 t plus 6 EndFraction commaN(t)=0.7t+8007t+6, 

t greater than or equals 15 commat≥15, 

gives the body concentration​ N(t), in parts per​ million, of a certain dosage of medication after time​ t, in hours.

​a) What does​ N(t) approach as t goes to 

infinity∞​?

​b) Explain the meaning of the answer to 

part a right parenthesispart a) 

in terms of the application.

...

Question content area right

Part 1

​N(t) approaches​ ___ as t goes to 

infinity∞.

A.

0

B.133.33

133.33133.33

C.0.1

0.10.1

D.infinity

infinity

Let's address the problem step by step.

### Part a) What does \( N(t) \) approach as \( t \to \infty \)?

The function given is:

\[
N(t) = \frac{0.7t + 800}{7t + 6}
\]

To find what \( N(t) \) approaches as \( t \) tends to infinity, we need to analyze the behavior of this function for very large values of \( t \). As \( t \) becomes very large, the terms with \( t \) in them will dominate over the constants (800 and 6), so we can simplify the function by focusing on the leading terms:

\[
N(t) \approx \frac{0.7t}{7t} = \frac{0.7}{7} = 0.1
\]

Thus, as \( t \to \infty \), \( N(t) \to 0.1 \).

The correct answer is **C: 0.1**.

---

### Part b) Interpretation of the result:

The result \( N(t) \to 0.1 \) means that as time passes, the concentration of the medication in the body approaches 0.1 parts per million. This suggests that after a long time, the dosage stabilizes at a very low concentration, indicating that the medication is being absorbed or eliminated from the body, leaving only a small residual amount.

---

Would you like more details or have any questions? 

Here are five related questions to expand your understanding:

1. How would the function behave for values of \( t \) less than 15?
2. What is the initial concentration of the medication at \( t = 15 \)?
3. How would the concentration change if the numerator and denominator coefficients were different?
4. What would happen if the medication had a higher concentration limit as \( t \to \infty \)?
5. Can this model be used to predict the half-life of the medication in the body?

**Tip:** When analyzing rational functions, always check the degree of the numerator and denominator to determine the function's end behavior.

The function N(t) = (0.7t + 800) / (7t + 6), t ≥ 15, gives the body concentration N(t), in parts per million, of a certain dosage of medication after time t, in hours. a) What does N(t) approach as t goes to infinity? b) Explain the meaning of the answer to part a) in terms of the application.

Learn how to analyze the end behavior of the rational function N(t) = (0.7t + 800) / (7t + 6) that models medication concentration over time. This problem explores the limit of N(t) as time approaches infinity and provides insights into how the dosage stabilizes in the body. Suitable for Grades 10-12 students studying limits and rational functions.

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