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

Question 3

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. | ... 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 during which year the height of the tree will increase by approximately 11.76 mm, we first find the common ratio $r$ of the geometric sequence.

Calculating $r$ :

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

Next, we use the formula for the $n^{th}$ term of a geometric sequence: $T_n = ar^{n-1}$ where:

$a = 100$ (first term)
$r = \frac{7}{10}$
$T_n = 11.76$ mm

We set up the equation: $11.76 = 100 \left( \frac{7}{10} \right)^{n-1}$

Solving for $n$ :

Divide both sides by 100: $\left( \frac{7}{10} \right)^{n-1} = \frac{11.76}{100}$ $\left( \frac{7}{10} \right)^{n-1} = 0.1176$
Take logarithms: $(n-1) \log\left(\frac{7}{10}\right) = \log(0.1176)$
Solve for $n$ : $n - 1 = \frac{\log(0.1176)}{\log\left(\frac{7}{10}\right)}$ $n = \frac{\log(0.1176)}{\log\left(\frac{7}{10}\right)} + 1$

After performing the calculations, we find that during the $7^{th}$ year, the increase will be approximately 11.76 mm.

Step 2

Determine a formula for $h(n)$

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Answer

To find the formula for the height of the tree, we start with the height after $n$ years:

The initial height is $h(0) = 130 mm$ , so the height after $n$ years can be expressed as: $h(n) = 130 + (100 + 70 + 49 + \ldots ) \text{ for } n \text{ terms}$

Using the geometric series sum formula: $S_n = a \frac{1 - r^n}{1 - r}$ where $a = 100$ and $r = \frac{7}{10}$ , we have: $h(n) = 130 + 100 + \frac{100(1 - (\frac{7}{10})^n)}{1 - \frac{7}{10}}$

Simplifying this gives a formula for $h(n)$ : $h(n) = 130 + 100 + \frac{100(1 - (\frac{7}{10})^n)}{0.3}$

Step 3

What height will the tree eventually reach?

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Answer

The eventual height of the tree can be determined by considering the limit of the height as $n$ approaches infinity.

As $n$ approaches infinity, $(\frac{7}{10})^n$ approaches 0. Therefore, the eventual height is: $h(\infty) = 130 + \frac{100}{0.3}$ Calculating this gives: $h(\infty) = 130 + 333.33 = 463.33 \text{ mm}$ Thus, 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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