Math Problem Statement
- 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
Solution
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:
- (growth in the first year).
- Each year, the growth is 85% of the previous year's growth, meaning the growth in year is given by: where is the growth in the -th year.
To find the growth in the fourth year:
Substituting the values:
Now, let's calculate this.
Rounding to two decimal places, the growth in the fourth year is approximately:
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:
- What would be the growth in the fifth year using the same formula?
- How much total growth does the tree experience over the first four years?
- What is the percentage decrease in growth between the first and fourth years?
- How would the total growth change if the decay rate was 90% instead of 85%?
- 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.
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Math Problem Analysis
Mathematical Concepts
Exponential Decay
Geometric Sequences
Formulas
g_n = g_1 * (0.85)^(n-1)
Theorems
Geometric Sequence Theorem
Suitable Grade Level
Grades 9-10
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