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Policeman Travel

by Andrew
(New York)













































A procession of cars, five miles long, drives at a constant speed for twelve miles and then stops. A policeman starts at the end of the procession as it begins. He rides at a constant speed to the front. Instantly, he turns around (magically he occupies the same space, but is facing the opposite direction), and rides to the back of the procession. He arrives at the moment the cars stop. How far did the policeman travel for the entire trip? (The answer is not 22.)

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Jan 03, 2012
Policeman Travel - Word Problem
by: Staff

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Part II

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DERIVATION of P = 3x/2

Total time = t

t = t₁ + t₂

when the policeman travels from the starting point (at the end of the procession) to the front of the procession, he travels 5 miles (the length of the procession) plus the distance the procession travels in time t₁


t₁ * P = 5 + (t₁ * x)

t₁ * P - (t₁ * x)= 5

t₁ * (P - x)= 5

t₁ = 5/(P - x)


(after turning around) the time it took for the policeman to travel from the front of the procession to the end of the procession is 5 miles minus the distance the procession travels in time t₂

t₂ * P = 5 - (t₂ * x)

t₂ * P + (t₂ * x) = 5

t₂ = 5/(P + x)


Solving for P

t = t₁ + t₂


substitute 12/x for t, 5/(P - x) for t₁,and 5/(P + x) for t₂

12/x = 5/(P - x) + 5/(P + x)


Multiply each side of the equation by (P-x)*(P+x)*x

(12/x)*(P-x)*(P+x)*x = [5/(P - x) + 5/(P + x)]*(P-x)*(P+x)*x

12(P² - x²) = 5*(P+x)*x + 5*(P-x)*x

12(P² - x²) = 5x*(P+x) + 5x*(P-x)

12(P² - x²) = 5Px + 5x² + 5Px - 5x²

12P² - 12x² = 5Px + 5Px + 5x² - 5x²

12P² - 12x² = 5Px + 5Px + 0

12P² - 12x² = 5Px + 5Px

12P² - 12x² = 10Px


Subtract 10Px from each side of the equation

12P² - 12x² - 10Px = 10Px - 10Px

12P² - 12x² - 10Px = 0

12P² - 10Px - 12x² = 0


Divide each side of the equation by 2

(12P² - 10Px - 12x²)/2 = 0/2

6P² - 5Px - 6x² = 0


Factor the left side of the equation

(2P - 3x)(3P +2x) = 0


Find the solution P

There are two possibilities:

1st possibility

(2P - 3x)*(3P +2x)/(3P +2x) = 0/(3P +2x)

(2P - 3x)*(3P +2x)/(3P +2x) = 0

(2P - 3x)*(1) = 0

(2P - 3x) = 0

2P - 3x = 0

2P - 3x + 3x = 0 + 3x

2P = 3x

2P/2 = 3x/2

P = 3x/2


2nd possibility

(2P - 3x)*(3P +2x)/( 2P - 3x) = 0/(2P - 3x)

(3P +2x)*(2P - 3x/( 2P - 3x) = 0

(3P +2x)*(1) = 0

(3P +2x) = 0

3P +2x = 0

3P + 2x - 2x = 0 - 2x

3P = -2x

3P/3 = -2x/3

P = -2x/3, this solution is INVALID since P must be a positive speed


In conclusion, there is only one valid solution for speed P

P = 3x/2

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Thanks for writing.

Staff
www.solving-math-problems.com


Jan 03, 2012
Policeman Travel - Word Problem
by: Staff


Part I

Question:

by Andrew
(New York)


A procession of cars, five miles long, drives at a constant speed for twelve miles and then stops. A policeman starts at the end of the procession as it begins. He rides at a constant speed to the front. Instantly, he turns around (magically he occupies the same space, but is facing the opposite direction), and rides to the back of the procession. He arrives at the moment the cars stop. How far did the policeman travel for the entire trip? (The answer is not 22.)


Answer:

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Definitions:

P = speed of policeman

x = speed of procession

t = Time it took for the entire procession to travel 12 miles

t₁ = Time it took for the policeman to travel from the starting point (at the end of the procession) to the front of the procession

t₂ = (after turning around) Time it took for the policeman to travel from the front of the procession to the end of the procession

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The LOGIC behind the SOLUTION:

The solution is based on a simple and straightforward principle:

The DISTANCE traveled by the policeman is EQUALS the SPEED the policeman travels MULTIPLIED BY the TIME. (The direction the policeman travels is not important)


Distance = Pt

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The SOLUTION:

t = 12/x
(the derivation of t = 12/x is shown below)


P = 3x/2
(the derivation of P = 3x/2 is shown below)


Distance = Pt

Distance = (3x/2)* 12/x

Distance = (3x*12)/(2*x)

Distance = [(3*12)/2]*(x/x)

Distance = [(3*12)/2]*(1)

Distance = (3*12)/2

Distance = 36/2

Distance = 18


>>>>>>>>>> The FINAL ANSWER is: 18 miles


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DERIVATION of t = 12/x


since the procession travels 12 miles in time t

t*x = 12


therefore, the time it takes for the procession to travel 12 miles is:

t = 12/x


t = 12/x is also the time it takes the motorcycle to travel from the end of the procession, to the front of the procession, and then back to the end of the procession

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