Step-by-step guide to solving a rate of change problem involving Daniel's height increase over time.

Problem Statement:

Daniel was 146 cm tall in 1999 and grew to be 176 cm by the year 2006. We need to determine Daniel's rate of growth over this period.

Solution:

#### Step 1: Find the length (change in height)
The change in Daniel's height is calculated by subtracting his initial height from his final height:
\[
\text{Change in height} = \text{Final height} - \text{Initial height}
\]
\[
\text{Change in height} = 176 \, \text{cm} - 146 \, \text{cm} = 30 \, \text{cm}
\]

#### Step 2: Find the time (duration of growth)
The duration of growth is calculated by subtracting the starting year from the ending year:
\[
\text{Duration of growth} = \text{Ending year} - \text{Starting year}
\]
\[
\text{Duration of growth} = 2006 - 1999 = 7 \, \text{years}
\]

#### Step 3: Find the rate of growth
The rate of growth is the change in height divided by the duration of growth:
\[
\text{Rate of growth} = \frac{\text{Change in height}}{\text{Duration of growth}}
\]
\[
\text{Rate of growth} = \frac{30 \, \text{cm}}{7 \, \text{years}} \approx 4.29 \, \text{cm/year}
\]

Final Answer:

\[
\boxed{4.29 \, \text{cm/year}}
\]

Explanation:

- Step 1 calculates the total increase in Daniel's height over the given period.
- Step 2 determines the number of years during which the growth occurred.
- Step 3 computes the average rate of growth by dividing the total change in height by the total time period. This gives us the average amount Daniel grew per year.

The result, approximately \(4.29 \, \text{cm/year}\), represents Daniel's average rate of growth from 1999 to 2006.