300 men were engaged to complete a job in a certain number of days. but due to another assignment fifteen men dropped the second day, fifteen more men dropped the third days and so on .if takes fifteen more days to finish the work now.find the number of days in which the work was completed. please hellllllppppp.
The number of man-hours that were used were
300 + 285 + 270 + 255 + ... + 15 + 0
Since we can't count the + 0 , (nobody worked that day) ,
we need the number of terms of
300+285 + ... + 15
the left side is an AS where a = 300 , d = -15 , n = ?
when the last term is 15
15 = a + d(n-1)
15 = 300 - 15(n-1)
15 = 300-15m + 15
15n = 300
n = 20
Reiny's solution just tells us the max number of days not how many days were actually used.
We know that 15 more days were used than originally planned with the full 300.
300d= the sum of the progression
Let's break down the problem step by step.
Step 1: Understand the initial situation
Initially, there were 300 men engaged to complete the job.
Step 2: Analyze the impact of men dropping out
On the second day, 15 men dropped out, leaving 300 - 15 = 285 men.
On the third day, another 15 men dropped out, leaving 285 - 15 = 270 men.
Following this pattern, we can see that the number of men remaining each day decreases by 15.
Step 3: Determine the number of men remaining on the last day
Since it takes 15 more days to finish the job, we need to calculate the total number of men remaining on the last day.
Let's assume that there are still "x" number of days remaining.
On the last day, there would be 300 - 15x men remaining.
Step 4: Find the total work done per day
Since the work rate of the men is constant, we can assume that each day they do 1/300th of the total work.
Step 5: Set up an equation
Now, let's form an equation based on the information provided. The total work remaining is equal to the work done each day multiplied by the number of days remaining.
(1/300) * (300 - 15x) * x = 15
Step 6: Solve the equation
To find the number of days (x), we need to solve the equation.
(1/300) * (300 - 15x) * x = 15
Divide both sides by 15:
(1/300) * (300 - 15x) * x = 1
Multiply both sides by 300 to eliminate the denominator:
(300 - 15x) * x = 300
Expand and rearrange the equation:
300x - 15x^2 = 300
Rearrange the equation:
15x^2 - 300x + 300 = 0
Divide both sides by 15:
x^2 - 20x + 20 = 0
Apply the quadratic formula to find the value of x:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 1, b = -20, and c = 20. Plugging these values into the formula, we get:
x = (-(-20) ± √((-20)^2 - 4 * 1 * 20)) / (2 * 1)
x = (20 ± √(400 - 80)) / 2
x = (20 ± √320) / 2
x = (20 ± 4√5) / 2
x = 10 ± 2√5
Since the number of days cannot be negative, we take the positive value:
x ≈ 10 + 2√5
Step 7: Calculate the number of days the work takes to complete
The number of days in which the work was completed would be approximately 10 + 2√5.
Note: The value obtained is an approximate value, as the exact calculation of the square root of 5 can be quite complex.
To determine the number of days it initially took to complete the work, we can start by subtracting the additional 15 days mentioned in the question.
Let's represent the number of days it initially took to complete the work by 'x'. So, after subtracting the additional 15 days, the new number of days to complete the work is 'x - 15'.
According to the given information, 300 men were engaged to complete the job initially. But, each day, starting from the second day, 15 men dropped out.
So, after the first day, the number of men working on the job would be 300 - 15 = 285.
After the second day, it would be 285 - 15 = 270.
After the third day, it would be 270 - 15 = 255.
This pattern continues until the work is completed after 'x' days. At this point, the number of men working on the job would be 300 - (15 * (x-1)).
Now, we can set up an equation based on the information provided:
300 - (15 * (x-1)) = 0
By solving this equation, we can find the value of 'x'.
300 - 15x + 15 = 0 (using distributive property)
-15x + 315 = 0
-15x = -315
Divide both sides by -15:
x = 21
Therefore, it initially took 21 days to complete the work before the 15 men dropped out each day.