How Tall Will the Tree Be?

68. How Tall Will the Tree Be?

A sapling was 2 m tall when it was planted. The tree increases its height by the same fixed amount each year.

Seven years after planting, the tree was one-eighth taller than it had been six years after planting.

How tall will the tree be ten years after it was planted?

Added 20 January 2012

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Hint:

Let g be the tree’s yearly growth (in metres). Express the heights after 6 and 7 years in terms of g and set up the given ratio.

Solution:

Let the constant yearly growth be g metres.

Height after 6 years: 2 + 6g
Height after 7 years: 2 + 7g

The statement “7 years height is one-eighth taller than 6 years height” means

(2 + 7g) = (2 + 6g) \(1 + \tfrac{1}{8}) = \tfrac{9}{8}(2 + 6g).

Multiply both sides by 8:

8(2 + 7g) = 9(2 + 6g)
16 + 56g = 18 + 54g

Rearrange:

56g – 54g = 18 – 16
2g = 2
g = 1 m per year.

Height after 10 years:

2 + 10g = 2 + 10(1) = 12 m.

Answer: 12 metres.

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