A pile driver pounds a steel column into the ground. On the first drive, the column is pounded 1.65 metres into the ground, and on each successive drive it moves 80% as far as it did on the previous drive. The least number of drives required to drive the column a total of 8.2 metres into the ground is _____.
so you want
1.65 + 1.65(.8) + 1.65(.8)^2 + .. = 8.2
you could just keep adding until you reach 8.2 but that would be less elegant than using a geometric series
a = 1.65
r = .8
Sum(n) = 8.2
a(1 - r^n)/(1-r) = sum(n)
1.65(1 - .8^n)/(1-.8) = 8.2
(1-.8^n)/.2 = 8.2/1.65
1-.8^n = 8.2/1.65*.2
.8^n = 1 - 8.2/1.65*.2
n log .8 = log [1 - 8.2/1.65*.2]
n = log [1- 8.2/1.65*.2] / log .8 = 22.8
22 drives will not do it, so they will need 23 pile drivings
check:
Sum(22) = 1.65 (1-.8^22)/.2 = 8.189
sum(23) = 1.65(1-.8^23)/.2 = 8.2013
my answer is correct.
To find the least number of drives required to drive the column a total of 8.2 meters into the ground, we can follow these steps:
Step 1: On the first drive, the column is pounded 1.65 meters into the ground.
Step 2: On each successive drive, it moves 80% as far as it did on the previous drive.
Let's calculate the distances covered in each drive:
Drive 1: 1.65 meters
Drive 2: 1.65 meters * 0.8 = 1.32 meters
Drive 3: 1.32 meters * 0.8 = 1.056 meters
Drive 4: 1.056 meters * 0.8 = 0.8448 meters
Drive 5: 0.8448 meters * 0.8 = 0.67584 meters
Now, let's add up the distances covered in the first five drives:
1.65 + 1.32 + 1.056 + 0.8448 + 0.67584 = 5.54664 meters
Since the column needs to be driven a total of 8.2 meters into the ground, we need to calculate how many additional drives are needed:
8.2 - 5.54664 = 2.65336 meters
Since each drive covers 80% of the previous drive's distance, the remaining distance of 2.65336 meters will be covered in less than one additional drive. Therefore, the least number of drives required to drive the column a total of 8.2 meters into the ground is 5 drives.
To find the least number of drives required to drive the column a total of 8.2 meters into the ground, we can start by determining the distance the column moves on each drive.
Given that the column moves 1.65 meters on the first drive, and on each successive drive it moves 80% as far as it did on the previous drive, we can use the formula:
Distance on nth drive = Distance on (n-1)th drive * 0.8
Let's calculate the distances on each drive:
1st drive: 1.65 meters
2nd drive: 1.65 * 0.8 = 1.32 meters
3rd drive: 1.32 * 0.8 = 1.056 meters
4th drive: 1.056 * 0.8 = 0.845 meters
5th drive: 0.845 * 0.8 = 0.676 meters
The column moves a total distance of 1.65 + 1.32 + 1.056 + 0.845 + 0.676 = 5.547 meters after 5 drives.
To reach a total distance of 8.2 meters, we need to find the least number of additional drives required. We can continue using the formula to find the distances on the next drives until we reach or surpass 8.2 meters.
6th drive: 0.676 * 0.8 = 0.541 meters
7th drive: 0.541 * 0.8 = 0.433 meters
8th drive: 0.433 * 0.8 = 0.345 meters
9th drive: 0.345 * 0.8 = 0.276 meters
After 9 drives, the column would have moved a total distance of 5.547 + 0.541 + 0.433 + 0.345 + 0.276 = 7.142 meters.
To reach a total distance of 8.2 meters, we need at least one more drive. Adding the distance moved on the 10th drive:
10th drive: 0.276 * 0.8 = 0.221 meters
After 10 drives, the column would have moved a total distance of 7.142 + 0.221 = 7.363 meters.
Therefore, the least number of drives required to drive the column a total of 8.2 meters into the ground is 10 drives.