SOLUTION: two technicians can complete a mailing in 3 hours when working together. alone, one can complete the mailing 2 hours faster than the other. how long will it take each person to com

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Question 206942: two technicians can complete a mailing in 3 hours when working together. alone, one can complete the mailing 2 hours faster than the other. how long will it take each person to complete the mailing alone? compute answer to two decimal places.
Found 2 solutions by nerdybill, Theo:
Answer by nerdybill(7384) About Me  (Show Source):
You can put this solution on YOUR website!
two technicians can complete a mailing in 3 hours when working together. alone, one can complete the mailing 2 hours faster than the other. how long will it take each person to complete the mailing alone? compute answer to two decimal places.
.
Let x = time it takes to complete mailing by slower tech
then
x-2 = time it takes by faster tech
.
3(1/x + 1/(x-2)) = 1
3/x + 3/(x-2) = 1
Multiply both sides by a common denominator x(x-2):
3(x-2) + 3x = x(x-2)
3x-6 + 3x = x^2-2x
6x-6 = x^2-2x
-6 = x^2-8x
0 = x^2-8x+6
Solve using the quadratic equation yields:
x = {7.16, 0.84}
We can throw out the 0.84 solution because it will give a negative answer for the faster tech leaving us with:
x = 7.16 hours for the slower tech
x-2 = 7.16-2 = 5.16 hours for the faster tech
.
Details of quadratic follows:
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B-8x%2B6+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-8%29%5E2-4%2A1%2A6=40.

Discriminant d=40 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--8%2B-sqrt%28+40+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-8%29%2Bsqrt%28+40+%29%29%2F2%5C1+=+7.16227766016838
x%5B2%5D+=+%28-%28-8%29-sqrt%28+40+%29%29%2F2%5C1+=+0.83772233983162

Quadratic expression 1x%5E2%2B-8x%2B6 can be factored:
1x%5E2%2B-8x%2B6+=+1%28x-7.16227766016838%29%2A%28x-0.83772233983162%29
Again, the answer is: 7.16227766016838, 0.83772233983162. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-8%2Ax%2B6+%29

Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
let x = rate of first technician
let y = rate of second technician.
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working together they can complete a mailing in 3 hours.
(x + y) * 3 = 1
(x + y) is their combined rate.
3 = number of hours
1 = units produced.
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they produced 1 unit of mailing in 3 hours working together.
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let the first technician be the faster mailer.
the first technician works at a rate of x mailings per hour.
the second technician works at a rate of y mailings per hour.
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alone, the first technician can complete the mailing 2 hours faster than the other.
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let h = amount of time in hours it takes for the first technician to complete the mailing.
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x * h = 1 mailing
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since the first technician can complete the mailing in 2 hours less than the second technician, then (h+2) represents the time it takes the second technician to complete the mailing.
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y * (h+2) = 1 mailing
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since x * h = 1, then x = (1/h) which looks like %281%2Fh%29
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since y * (h+2) = 1, then y = (1/(h+2)) which looks like %281%2F%28h%2B2%29%29
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formula of (x+y)*3 = 1 becomes ((1/h) + (1/(h+2))) * 3 = 1
this looks like %28%281%2Fh%29+%2B+%281%2F%28h%2B2%29%29%29+%2A+3+=+1
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multiply both sides of this equation by (h)*(h+2) to get ((h+2) + (h)) * 3 = (h)*(h+2)
this looks like %28%28h%2B2%29+%2B+%28h%29%29+%2A+3+=+%28h%29%2A%28h%2B2%29
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this becomes 6h + 6 = h^2 + 2h which looks like 6h+%2B+6+=+h%5E2+%2B+2h
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this eventually results in h^2 - 4h - 6 = 0 which looks like h%5E2+-+4h+-+6+=+0
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using the quadratic formula, the roots are either:
h = 5.16227766 or h = -1.16227766
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since h can't be negative, the only possible answer is h = 5.16227766
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if h = 5.16227766, then x = 1/5.16227766 = .193712943 and y = 1/7.16227766 = .13962039
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to test this out, we substitute for x and h in the equations we created that contain h in them.
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first equation is %28%281%2Fh%29+%2B+%281%2F%28h%2B2%29%29%29+%2A+3+=+1
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this becomes %28%281%2F5.16227766%29+%2B+%281%2F7.16227766%29%29+%2A+3+=+1
which results in 1 = 1 proving the value of h is correct.
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second equation is x * h = 1.
x = .193712943
h = 5.16227766
x * h = 1 proving the values for x and h are good.
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third equation is y * (h+2) = 1.
y = .13962039
h + 2 = 7.16227766
y * (h+2) = 1 proving the values for y and (h+2) are good.
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we started off with %28x%2By%29+=+1
we then got %28x%2Ah%29+=+1
we then got %28y%2A%28h%2B2%29%29+=+1
we then got x+=+%281%2Fh%29
we then got y+=+%281%2F%28h%2B2%29%29
we then substituted in %28x%2By%29+=+1 to get %28%281%2Fh%29+%2B+%281%2F%28h%2B2%29%29%29+%2A+3+=+1
we then solved for h.
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once we solved for h we were then able to complete the problem.
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