Each year when the sunflower paddock is weeded, only half the previous weight of weed is dug out. In the first year, 6 tonnes of weed is dug out.
i How much is dug out in the 10th year?
ii What is the total dug out over 10 years (correct to four significant figures)?
To find the amount of weed dug out in each year, we can use the information that only half of the previous year's weight of weed is removed.
i) To find the amount of weed dug out in the 10th year, we need to determine how much weed is left after each year and then calculate the weed dug out in the 10th year.
Year 1: 6 tonnes of weed dug out
Year 2: Half of the previous year's weight (6/2 = 3 tonnes) dug out
Year 3: Half of the previous year's weight (3/2 = 1.5 tonnes) dug out
...
Year 10: Half of the previous year's weight (x/2) dug out
To find the amount of weed dug out in the 10th year, let's work backward from the given information for Year 1:
Year 1: 6 tonnes (given)
Year 2: 3 tonnes (half of the previous year's weight)
Year 3: 1.5 tonnes (half of the previous year's weight)
...
Year 10: x tonnes (half of the previous year's weight)
Since we are looking for the 10th year, we need to keep dividing the previous year's weight by 2 nine times (since the 1st year is already accounted for).
Year 10: (6/2) / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2
= (6/2^9) tonnes
To calculate this, we can simplify using the formula for halving a number multiple times, which is given by:
x / 2^n
where x is the initial weight (6 tonnes), and n is the number of times we halve (9 in this case).
Calculating Year 10:
= 6 / 2^9
= 6 / 512
= 0.0117174 tonnes (approx.)
Therefore, approximately 0.0117 tonnes (or 11.7 kilograms) of weed is dug out in the 10th year.
ii) To find the total amount of weed dug out over 10 years, we need to sum up the weed dug out in each year:
Total = Year 1 + Year 2 + Year 3 + ... + Year 10
Since the pattern is halving the previous year's weight, we can use the formula for the sum of a geometric series:
Total = a * (1 - r^n) / (1 - r)
where a is the first term (6 tonnes), r is the common ratio (1/2), and n is the number of terms (10 years).
Calculating the total:
Total = 6 * (1 - (1/2)^10) / (1 - 1/2)
≈ 6 * 0.9990234375 / (1/2)
≈ 5.994140625 / 0.5
≈ 11.98828125 tonnes (approx.)
Therefore, the total dug out over 10 years, rounded to four significant figures, is approximately 11.99 tonnes.