Question

tree growth
an arborist wants to graph the growth of a tree over its 50-year life span
create a line graph to show the tree’s growth
table with age (years) and height (meters): 10, 6.6; 20, 14.8; 30, r.8 (likely typo, e.g., 19.8); 40, 23.7; 50, 25
graph grid with x and y axes
use the graph to answer the following questions.

1. during which time span did the tree have the most growth?
2. during which time span did the tree have the least growth?
3. using the graph, estimate the height of the tree at 25 years
4. using the graph, estimate the height of the tree at 35 years

Explanation:

Step 1: Prepare the Graph Axes

• X - axis (Independent Variable): Represent "Age (years)". Label the x - axis with intervals: 10, 20, 30, 40, 50.
• Y - axis (Dependent Variable): Represent "Height (meters)". Determine a suitable scale. Since the heights range from 6.6 to 25 meters, we can use a scale where each grid square represents, for example, 2 meters. So the y - axis will have labels like 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26.

Step 2: Plot the Data Points

• For age = 10 years, height = 6.6 meters. Find the point (10, 6.6) on the graph (x = 10, y≈6.6).
• For age = 20 years, height = 14.8 meters. Find the point (20, 14.8) (x = 20, y≈14.8).
• For age = 30 years, height = 19.8 meters. Find the point (30, 19.8) (x = 30, y≈19.8).
• For age = 40 years, height = 23.7 meters. Find the point (40, 23.7) (x = 40, y≈23.7).
• For age = 50 years, height = 25 meters. Find the point (50, 25) (x = 50, y = 25).

Step 3: Draw the Line Graph

Connect the plotted points (10, 6.6), (20, 14.8), (30, 19.8), (40, 23.7), and (50, 25) with a straight line (or a line that best fits the trend) to form the line graph.

Answer:

ing the Questions:

1. Time span with most growth

• Calculate the growth (change in height) for each time span:
• 10 - 20 years: \(14.8 - 6.6=8.2\) meters.
• 20 - 30 years: \(19.8 - 14.8 = 5\) meters.
• 30 - 40 years: \(23.7 - 19.8=3.9\) meters.
• 40 - 50 years: \(25 - 23.7 = 1.3\) meters.
• The largest growth is 8.2 meters, which occurs between 10 - 20 years.

2. Time span with least growth

• From the growth calculations above:
• The smallest growth is 1.3 meters, which occurs between 40 - 50 years.

3. Estimate height at 25 years

• The time between 20 and 30 years. The height at 20 is 14.8 m and at 30 is 19.8 m. The mid - point of 20 and 30 is 25. The average rate of growth between 20 - 30 is \(\frac{19.8 - 14.8}{30 - 20}=\frac{5}{10} = 0.5\) m/year.
• From 20 to 25 is 5 years. So the height at 25 is \(14.8+5\times0.5=14.8 + 2.5=17.3\) meters (or by looking at the graph, since the line between (20,14.8) and (30,19.8) is linear, the value at x = 25 is the average of 14.8 and 19.8: \(\frac{14.8 + 19.8}{2}=\frac{34.6}{2}=17.3\) meters).

4. Estimate height at 35 years

• The time between 30 and 40 years. The height at 30 is 19.8 m and at 40 is 23.7 m. The mid - point of 30 and 40 is 35. The average rate of growth between 30 - 40 is \(\frac{23.7 - 19.8}{40 - 30}=\frac{3.9}{10}=0.39\) m/year.
• From 30 to 35 is 5 years. So the height at 35 is \(19.8+5\times0.39=19.8 + 1.95 = 21.75\) meters (or by taking the average of 19.8 and 23.7: \(\frac{19.8+23.7}{2}=\frac{43.5}{2}=21.75\) meters).

Final Answers:

1. The tree had the most growth between \(\boldsymbol{10 - 20}\) years.
2. The tree had the least growth between \(\boldsymbol{40 - 50}\) years.
3. The estimated height at 25 years is \(\boldsymbol{17.3}\) meters (approximate, may vary slightly based on graph drawing).
4. The estimated height at 35 years is \(\boldsymbol{21.75}\) meters (approximate, may vary slightly based on graph drawing).