8. A tree grows 2.80 m during the first year since it was planted. During each subsequent year, the tree growth is 85% of its growth the previous year. a) Calculate, to 0.01 m, the growth of the tree in the fourth year

The problem describes a situation where the growth of the tree decreases by a fixed percentage each year after the first. This suggests an exponential decay model for the tree's growth over time.

Let's define the following:

- \( g_1 = 2.80 \, \text{m} \) (growth in the first year).
- Each year, the growth is 85% of the previous year's growth, meaning the growth in year \( n \) is given by:
  \[
  g_n = g_1 \times (0.85)^{n-1}
  \]
  where \( g_n \) is the growth in the \( n \)-th year.

### To find the growth in the fourth year:
\[
g_4 = g_1 \times (0.85)^{3}
\]
Substituting the values:
\[
g_4 = 2.80 \times (0.85)^3
\]

Now, let's calculate this.

\[
g_4 = 2.80 \times 0.614125 = 1.71955 \, \text{m}
\]

Rounding to two decimal places, the growth in the fourth year is approximately:

\[
g_4 \approx 1.72 \, \text{m}
\]

### Answer:
The growth of the tree in the fourth year is approximately **1.72 meters**.

Would you like further details or have any other questions?

Here are 5 related questions:
1. What would be the growth in the fifth year using the same formula?
2. How much total growth does the tree experience over the first four years?
3. What is the percentage decrease in growth between the first and fourth years?
4. How would the total growth change if the decay rate was 90% instead of 85%?
5. What is the long-term trend for the tree’s growth after many years?

**Tip:** The pattern of decay in this problem is modeled by a geometric sequence, which can be useful in many real-world applications like population decline or radioactive decay.

A tree grows 2.80 m during the first year since it was planted. During each subsequent year, the tree growth is 85% of its growth the previous year. a) Calculate, to 0.01 m, the growth of the tree in the fourth year.

This problem explores exponential decay and geometric sequences to calculate the growth of a tree that grows 2.80 m in the first year and decreases its growth by 85% each subsequent year. Learn how to apply the geometric sequence formula to determine the growth in the fourth year, approximately 1.72 meters. Suitable for students in Grades 9-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation