https://file.aiquickdraw.com/mathbot/file/1726537729721ri9gmob7.jpg

The problem displayed in the image involves finding an exponential function that models the growth of whooping cough cases, and calculating the average annual percent growth rate.

### Let’s break it down:

1. **Part (a)**:
   The exponential model has the form:
   \[
   w = w_0 \cdot r^t
   \]
   Where:
   - \( w \) is the number of cases,
   - \( w_0 \) is the initial number of cases (1248 in 1981),
   - \( r \) is the growth factor,
   - \( t \) is time in years since 1980.

   The general formula \( w = 1248 \cdot 1.141^t \) is proposed in the problem, but it appears to have an incorrect verification mark.

2. **Part (b)**:
   This part asks for the annual percent growth rate based on the function from part (a). 

   The growth factor is \( r = 1.141 \), and to convert this to a percent growth rate:
   \[
   \text{Growth rate} = (r - 1) \times 100 = (1.141 - 1) \times 100 = 14.1\%
   \]
   So the annual growth rate is **14.1%**.

3. **Part (c)**:
   This question mentions a specific date (May 4, 2005) and asks if the number of cases more than doubled. To determine this, we would need to plug in \( t = 25 \) (since 2005 is 25 years after 1980) into the exponential model and check if the result is more than double the initial value of 1248.

---

Do you want me to go into more details on a specific part, or clarify further? Let me know!

### Related questions:
1. How do we find the correct exponential model if the provided one is incorrect?
2. What happens to the growth rate if we change the base growth factor?
3. How does exponential growth compare to linear growth over long periods?
4. Could external factors slow down the growth rate in real-world scenarios?
5. How does rounding affect the accuracy of mathematical models?

**Tip**: Always double-check exponential calculations, especially when applying them over long periods to avoid minor errors becoming significant.

Whooping cough was thought to have been almost wiped out by vaccinations. It is now known that the vaccination wears off, leading to an increase in the number of cases, w, from 1248 in 1981 to 25,827 in 2004. (a) With t in years since 1980, find an exponential function that fits this data. (b) What does your answer to part (a) give as the average annual percent growth rate of the number of cases? (c) On May 4, 2005, the Arizona Daily Star newspaper reported that the number of cases had more than doubled.

Explore the exponential growth model for whooping cough cases from 1981 to 2004 and calculate the average annual percent growth rate. This problem involves key algebraic concepts, exponential functions, and percent growth formulas, suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation