To determine the height to which the plant grew during the third year, we can use the information given in the question.
In the first year, the plant grew to a height of 80 mm. In the second year, the height increased by 30 mm, so the plant's height at the end of the second year would be:
80 mm + 30 mm = 110 mm
From the third year onwards, the plant's annual growth is equal to the growth of the previous year. So, to calculate the plant's height at the end of the third year, we can add the growth in the third year to the height at the end of the second year:
110 mm + 30 mm = 140 mm
Therefore, the plant grew to a height of 140 mm during the third year.
To calculate the maximum height that the plant will reach, we can continue this pattern of adding the growth from the previous year to the current height. We can use a formula to calculate the height of the plant after n years:
Height = 80 + 30 + 30^2 + 30^3 + ... + 30^(n-1)
This is a geometric sequence with a common ratio of 30/1 = 30, and a first term of 80. The formula for the sum of the first n terms of a geometric sequence is:
Sum = (first term * (1 - common ratio^n)) / (1 - common ratio)
Substituting the values from the question, we get:
Sum = (80 * (1 - 30^n)) / (1 - 30)
Simplifying this expression, we get:
Sum = (30^n - 1) / 29
To find the maximum height that the plant will reach, we need to find the value of n that makes the sum equal to or just less than the desired maximum height. Let's assume the maximum height is H mm. Then we can solve for n:
H = (30^n - 1) / 29
29H = 30^n - 1
30^n = 29H + 1
n = log(29H + 1) / log(30)
Using a calculator, we can find that n is approximately 4.316. We need to round up to the nearest whole number, since the plant can't grow to a fraction of a millimeter. Therefore, the maximum height that the plant will reach is:
Height = 80 + 30 + 30^2 + 30^3 + 30^4
Height = 80 + 30 + 900 + 27,000 + 810,000
Height = 837,010 mm
Therefore, the maximum height that the plant will reach is 837,010 mm, or approximately 837.01 cm, or 8.37 meters.