A gardener has the task of digging an area of 800 metres per square. On the first day he digs an area of 10 metres per square. On each successive day he digs an area of 1.2 times the area that he dug the previous day, until the day when the task is completed. Find the number of days needed to complete the task.
you have a GP with
a = 10
r = 1.2
So you want
10(1.2^n - 1)/(1.2 - 1) >= 800
Bsnsn
To find the number of days needed to complete the task, we can set up an equation.
Let's assume that the gardener completes the task in n days.
On the first day, the gardener digs 10 square meters.
On the second day, the gardener digs 1.2 times the area dug on the previous day, which is 1.2 * 10 = 12 square meters.
On the third day, the gardener digs 1.2 times the area dug on the previous day, which is 1.2 * 12 = 14.4 square meters.
We can observe that on each day, the area being dug increases by a factor of 1.2.
So, the area dug on the nth day will be equal to 10 * (1.2)^(n-1).
The total area of 800 square meters should be equal to the sum of all the areas dug up to that day.
Therefore, we can set up the following equation:
10 + 12 + 14.4 + ... + 10 * (1.2)^(n-1) = 800
To solve this equation, we can use a geometric series formula to simplify it.
The sum of a geometric series is given by the formula:
S = a * (1 - r^n) / (1 - r),
where S is the sum, a is the first term, r is the common ratio, and n is the number of terms.
In this case, a = 10, r = 1.2, and S = 800.
800 = 10 * (1 - (1.2)^n) / (1 - 1.2)
800 * (1 - 1.2) = 10 * (1.2)^n - 10
-160 = 10 * (1.2)^n - 10
10 * (1.2)^n = 10 - 160
10 * (1.2)^n = -150
(1.2)^n = -150 / 10
(1.2)^n = -15
Since a negative value raised to any power will never be equal to a positive number, there is no solution to this equation. Therefore, the task cannot be completed.
So, there is no number of days that will complete the task.
To find the number of days needed to complete the task, we need to determine how many days it takes for the gardener to dig an area of 800 square meters.
Let's start by calculating how many square meters the gardener digs each day.
On the first day, the gardener digs an area of 10 square meters.
To find out the area the gardener digs on the second day, we need to multiply the area dug on the previous day by 1.2. So, on the second day, the gardener digs an area of 10 * 1.2 = 12 square meters.
Similarly, on the third day, the gardener digs an area of 12 * 1.2 = 14.4 square meters.
We can see that the area dug each day follows a pattern of multiplying by 1.2.
To find the day when the task is completed, we need to sum up the areas dug each day until it reaches or exceeds 800 square meters.
Let's calculate the total area dug each day until the task is completed:
Day 1: 10 square meters
Day 2: 12 square meters
Day 3: 14.4 square meters
Day 4: 14.4 * 1.2 = 17.28 square meters
Day 5: 17.28 * 1.2 = 20.736 square meters
Day 6: 20.736 * 1.2 = 24.8832 square meters
Day 7: 24.8832 * 1.2 = 29.85984 square meters
Day 8: 29.85984 * 1.2 = 35.831808 square meters
Day 9: 35.831808 * 1.2 = 42.9981696 square meters
Day 10: 42.9981696 * 1.2 = 51.59780352 square meters
After 10 days, the gardener has dug an area of approximately 51.6 square meters, which is less than 800 square meters.
Now, let's continue calculating until we reach or exceed 800 square meters:
Day 11: 51.59780352 * 1.2 = 61.91736422 square meters
Day 12: 61.91736422 * 1.2 = 74.30083706 square meters
Day 13: 74.30083706 * 1.2 = 89.16100448 square meters
Day 14: 89.16100448 * 1.2 = 107.7932054 square meters
Day 15: 107.7932054 * 1.2 = 129.3518465 square meters
Day 16: 129.3518465 * 1.2 = 155.2222158 square meters
Day 17: 155.2222158 * 1.2 = 186.2666589 square meters
After 17 days, the gardener has dug an area of approximately 186.3 square meters, which is still less than 800 square meters.
Repeating this process, we can continue calculating until we reach or exceed 800 square meters.
Day 18: 186.2666589 * 1.2 = 223.5199907 square meters
Day 19: 223.5199907 * 1.2 = 268.2239888 square meters
Day 20: 268.2239888 * 1.2 = 321.8687866 square meters
Day 21: 321.8687866 * 1.2 = 386.2425439 square meters
Day 22: 386.2425439 * 1.2 = 463.4910527 square meters
Day 23: 463.4910527 * 1.2 = 556.1892632 square meters
Day 24: 556.1892632 * 1.2 = 667.4271158 square meters
Day 25: 667.4271158 * 1.2 = 800.912539 square meters
After 25 days, the gardener has dug an area of 800 square meters, completing the task.
Therefore, the number of days needed to complete the task is 25 days.