Chris bought a bonsai (miniature tree) at a nursery. When he bought the tree, its height was 130 mm. Thereafter the height of the tree increased, as shown below. INCREASE IN HEIGHT OF THE TREE PER YEAR During the first year 100 mm During the second year 70 mm During the third year 49 mm Chris noted that the sequence of height increases, namely 100; 70; 49 ..., was geometric. During which year will the height of the tree increase by approximately 11,76 mm? Chris plots a graph to represent the height h(n) of the tree (in mm) n years after he bought it. Determine a formula for h(n). What height will the tree eventually reach?

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Chris bought a bonsai (miniature tree) at a nursery - NSC Mathematics - Question 3 - 2016 - Paper 1

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Chris bought a bonsai (miniature tree) at a nursery. When he bought the tree, its height was 130 mm. Thereafter the height of the tree increased, as shown below. IN... show full transcript

Worked Solution & Example Answer:Chris bought a bonsai (miniature tree) at a nursery - NSC Mathematics - Question 3 - 2016 - Paper 1

Step 1

During which year will the height of the tree increase by approximately 11,76 mm?

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Answer

To determine when the height increase will be approximately 11,76 mm, we start by identifying the common ratio of the geometric sequence:

$r = \frac{70}{100} = \frac{7}{10}$

The general formula for the height increase at year n is given by:

$T_n = ar^{n-1}$

Where:

a = the first term (100 mm)
r = common ratio

Set up the equation:

$11,76 = 100 \left(\frac{7}{10}\right)^{n-1}$

To find n, rearranging gives:

$\left(\frac{7}{10}\right)^{n-1} = \frac{11,76}{100} = 0,1176$

Taking the logarithm on both sides:

$(n-1) \log \left(\frac{7}{10}\right) = \log(0,1176)$

Thus,

$n - 1 = \frac{\log(0,1176)}{\log(0,7)}$

Calculating gives:

$n \approx 6$

Therefore, the height will increase by approximately 11,76 mm during the 7th year.

Step 2

Determine a formula for h(n).

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Answer

The height h(n) of the tree after n years can be expressed using the geometric sequence formula:

$h(n) = 130 + 100 + 70 + 49 + \ldots$

This is a sum of a geometric series:

Initial height: 130 mm
First year increase: 100 mm
Second year increase: 70 mm which can be expressed as:

$h(n) = 130 + \left(100 \cdot 0.7^{n-1}\right)$

Summing this gives:

$h(n) = 130 + \frac{100(1 - 0.7^n)}{1 - 0.7}$

That simplifies to:

$h(n) = 130 + \frac{100 \cdot 0.7^n}{0.3}$

Step 3

What height will the tree eventually reach?

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Answer

To find the eventual height of the tree, as n approaches infinity, the term involving the geometric sequence diminishes. The formula becomes:

$\lim_{n \to \infty} h(n) = 130 + \frac{100}{1 - 0.7}$

Therefore:

$h(n) = 130 + 100 \cdot 3.333... = 130 + 333.33 = 463.33$

The tree will eventually reach a height of approximately 463.33 mm.

Chris bought a bonsai (miniature tree) at a nursery - NSC Mathematics - Question 3 - 2016 - Paper 1

Worked Solution & Example Answer:Chris bought a bonsai (miniature tree) at a nursery - NSC Mathematics - Question 3 - 2016 - Paper 1

During which year will the height of the tree increase by approximately 11,76 mm?

Determine a formula for h(n).

What height will the tree eventually reach?

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