https://r2.erweima.ai/mathbot/image/1726530364650dvmfao6m.jpg

Let's break down the tasks in the image based on the question parts:

### (a) Find the average rate of change of the percent of the population living in urban areas between 1890 and 1990.

The formula for average rate of change is:

\[
\frac{f(b) - f(a)}{b - a}
\]

Using the given table, we have:
- In 1890, the percent of population in urban areas was 38.6.
- In 1990, it was 75.6.

So, applying the formula:

\[
\frac{75.6 - 38.6}{1990 - 1890} = \frac{37}{100} = 0.37 \text{ percent/year}.
\]

This matches the provided answer.

---

### (b) Estimate the rate at which this percent is increasing in 1990.

To estimate the rate of change at 1990, we use the surrounding years. Looking at the dataLet's break down the tasks in the image based on the question parts:

### (a) Find the average rate of change of the percent of the population living in urban areas between 1890 and 1990.

The formula for average rate of change is:

\[
\frac{f(b) - f(a)}{b - a}
\]

Using the given table, we have:
- In 1890, the percent of population in urban areas was 38.6.
- In 1990, it was 75.6.

So, applying the formula:

\[
\frac{75.6 - 38.6}{1990 - 1890} = \frac{37}{100} = 0.37 \text{ percent/year}.
\]

This matches the provided answer.

---

### (b) Estimate the rate at which this percent is increasing in 1990.

To estimate the rate of change at 1990, we use the surrounding years. Looking at the 
- In 1980, the percent is 73.5.
- In 2000, the percent is 79.8.

The rate of change between these two years is:

\[
\frac{79.8 - 73.5}{2000 - 1980} = \frac{6.3}{20} = 0.315 \text{ percent/year}.
\]

This suggests that the rate is close to \( 0.39 \), but the exact method may vary slightly.

---

### (c) Estimate the rate of change for 1830.

The values from the table are:
- In 1800, the percent is 6.6.
- In 1860, the percent is 17.1.

The rate of change between these two years is:

\[
\frac{17.1 - 6.6}{1860 - 1800} = \frac{10.5}{60} \approx 0.175 \text{ percent/year}.
\]

However, for 1830, we estimate between 1800 and 1860:
\[
\text{Estimated value: between 0.085 and 0.15 percent/year}.
\]

---

### (d) Is the function increasing or decreasing from 1800 to 1860?

From the data, the percentage increased from 6.6 in 1800 to 17.1 in 1860, meaning the function is **increasing**.

---

Let me know if you'd like further clarification or details.

### Related Questions:
1. What is the average rate of change between 1950 and 2000?
2. How do you calculate the average rate of change between two points?
3. How does the rate of change between 1830 and 1890 compare with later periods?
4. What does a higher average rate of change indicate about population trends?
5. How could this table be used to predict future urbanization rates?

**Tip:** Always compare nearby points for better estimations when the data table is provided in intervals.

Find the average rate of change of the percent of the population living in urban areas between 1890 and 1990, estimate the rate for 1990, and analyze the change for other years.

This problem involves calculating the average rate of change of urban population percentages between 1890 and 1990. Using the formula for rate of change, we also estimate the rate for the year 1990 and analyze the growth trend between 1800 and 1860.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation