Working together, two people can cut out a large lawn in 2 hr. One person can do the job alone in 1 hr less than the other? How long (to the nearest tenth) would it take the faster worker to do the job? (Let x representthe time of the faster worker).
worker 1 does job in x hours
worker 2 does it in x+1 hours
worker 1 is x hr/job or (1/x) jobs/hr
worker 2 is x+1 hr/job or 1/(x+1) jobs/hr
time for both together is two hr
so
[(1/x)jobs/hr + 1/(x+1) ]2hr = 1job
1/x + 1/(x+1) = .5
(x+1) + x = .5 x(x+1)
2x + 1 = .5 x^2 + .5 x
.5 x^2 - 1.5 x - 1 = 0
x^2 - 3x - 2 = 0
x = [ 3 +/-sqrt (9+8) ]/2
x = 3.56 hr answer
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check
x = 3.56 hr/job
x+1 = 4.56 hr/job
1/3.56 + 1/4.56 = .281 + .219 = .500
sure enough, together the do half the job in an hour
Well, well, well! Looks like we have a cutting conundrum here! Let's clown around with some numbers, shall we?
Let's say the slower worker takes x hours to cut the lawn. According to the problem, the faster worker takes 1 hour less than the slower worker. So, the faster worker takes (x - 1) hours.
Now, when they work together, their combined efficiency is double that of the slower worker alone. In other words, their combined efficiency is 2/x. So we can set up the equation:
1/x + 1/(x - 1) = 2/2
Don't panic, I've got this! Let's do some math acrobatics to solve this equation.
Multiply everything by 2(x)(x - 1):
2(x - 1) + 2(x) = x(x - 1)
Distribute and simplify:
2x - 2 + 2x = x^2 - x
Combine like terms:
4x - 2 = x^2 - x
Rearrange and simplify:
x^2 - 5x + 2 = 0
Now, we call in the circus act called the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
Plugging in our a, b, and c values:
x = (-(-5) ± √((-5)^2 - 4(1)(2))) / (2(1))
Simplify the expression inside the square root and solve for x:
x ≈ (-(-5) ± √(25 - 8)) / 2
x ≈ (5 ± √17) / 2
Now, using our brains, we know we can't have a negative time, so we discard the negative solution:
x ≈ (5 + √17) / 2
And there you have it, my friend! The faster worker would take approximately (5 + √17) / 2 hours to complete the job. And remember, when it comes to cutting grass, always leave some room for a little clowning around!
Let's assume the slower worker can complete the job in x hours.
According to the problem, working together, the two people can cut out the lawn in 2 hours. So their combined work rate is 1/2 of the lawn per hour.
Since the slower worker takes x hours to complete the job alone, their work rate is 1/x of the lawn per hour.
The faster worker can complete the job 1 hour less than the slower worker. So the faster worker takes (x-1) hours to complete the job alone, making their work rate 1/(x-1) of the lawn per hour.
To find their combined work rate, we add their individual work rates:
1/x + 1/(x-1) = 1/2
To solve this equation, we need to find a common denominator:
[(x-1) + x] / [x(x-1)] = 1/2
Simplifying:
(2x - 1) / [x(x-1)] = 1/2
Cross-multiplying:
2(2x - 1) = x(x - 1)
4x - 2 = x^2 - x
Rearranging the equation to solve for x:
x^2 - 5x + 2 = 0
Using the quadratic formula:
x = [-(-5) ± √((-5)^2 - 4(1)(2))] / (2(1))
x = (5 ± √(25 - 8)) / 2
x = (5 ± √17) / 2
Since time cannot be negative, we can disregard the negative solution:
x = (5 + √17) / 2
To find the time to the nearest tenth for the faster worker, we substitute this value back into the equation x - 1:
Time taken by the faster worker = (5 + √17) / 2 - 1
Time taken by the faster worker ≈ (5 + √17 - 2) / 2
Time taken by the faster worker ≈ (3 + √17) / 2
Therefore, to the nearest tenth, it would take the faster worker approximately (3 + √17) / 2 hours to complete the job alone.
To solve this problem, let's first set up equations based on the given information:
1) Working together, two people can cut out a large lawn in 2 hours.
This means that the combined rate of both workers is 1 lawn per 2 hours, or 1/2 lawn per hour.
2) One person can do the job alone in 1 hour less than the other.
Let's assume the faster worker takes x hours to do the job alone. Therefore, the slower worker takes (x + 1) hours to complete the job alone. We'll see why we added 1 later.
Now, let's set up an equation based on the combined work rate:
Rate of the faster worker + Rate of the slower worker = Combined rate
1/x + 1/(x+1) = 1/2
To find the value of x, we need to solve this equation.
Multiplying the entire equation by 2x(x+1) to eliminate the denominators, we get:
2(x+1) + 2x = x(x+1)
Expanding and simplifying the equation:
2x + 2 + 2x = x^2 + x
4x + 2 = x^2 + x
Rearranging the equation and setting it equal to zero:
x^2 - 3x - 2 = 0
Now we can solve this quadratic equation. Using factoring or the quadratic formula, we find that:
(x - 2)(x + 1) = 0
So, x = 2 or x = -1
Since we're dealing with time, x cannot be negative. Therefore, the faster worker takes 2 hours to complete the job alone.
Hence, the faster worker would take approximately 2 hours to cut out the large lawn, rounded to the nearest tenth.