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The rules for a race require that all runners start at A, touch any part of the 1200-meter wall, and stop at B. What is the number of meters in the minimum distance a participant must run? Express your answer to the nearest meter.

 

 Nov 6, 2018
 #1
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I found https://web2.0calc.com/questions/the-rules-for-a-race-require-that-all-runners-start but I can't really understand...?

 Nov 6, 2018
 #2
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That method works, but if you are looking for a much easier method here is one:

 

Reflect \(A\) and \(B\) \(180°\)down, and mark those new points as \(A'\) and \(B'\). You can see that running from \(A\) to \(B'\) is the same distance as a route that would fit in the requirments of the race. Now, we have to find the shortest route from \(A\) to \(B'\). The shortest route is a straight line. This straight line's length is \(1442\) m. 

 

- Daisy

 Nov 6, 2018
 #3
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See answer here: https://web2.0calc.com/questions/the-rules-for-a-race-require-that-all-runners-start
Solve for x over the real numbers:
x/sqrt(x^2 + 90000) + (x - 1200)/sqrt(x^2 - 2400 x + 1690000) = 0

Subtract (x - 1200)/sqrt(x^2 - 2400 x + 1690000) from both sides:
x/sqrt(x^2 + 90000) = -(x - 1200)/sqrt(x^2 - 2400 x + 1690000)

Cross multiply:
x sqrt(x^2 - 2400 x + 1690000) = (1200 - x) sqrt(x^2 + 90000)

Raise both sides to the power of two:
x^2 (x^2 - 2400 x + 1690000) = (1200 - x)^2 (x^2 + 90000)

Expand out terms of the left hand side:
x^4 - 2400 x^3 + 1690000 x^2 = (1200 - x)^2 (x^2 + 90000)

Expand out terms of the right hand side:
x^4 - 2400 x^3 + 1690000 x^2 = x^4 - 2400 x^3 + 1530000 x^2 - 216000000 x + 129600000000

Subtract x^4 - 2400 x^3 + 1530000 x^2 - 216000000 x + 129600000000 from both sides:
160000 x^2 + 216000000 x - 129600000000 = 0

The left hand side factors into a product with three terms:
160000 (x - 450) (x + 1800) = 0

Divide both sides by 160000:
(x - 450) (x + 1800) = 0

Split into two equations:
x - 450 = 0 or x + 1800 = 0

Add 450 to both sides:
x = 450 or x + 1800 = 0

Subtract 1800 from both sides:
x = 450 or x = -1800[DISCARD THIS ONE]


x/sqrt(x^2 + 90000) + (x - 1200)/sqrt(x^2 - 2400 x + 1690000) ⇒ 450/sqrt(90000 + 450^2) + (450 - 1200)/sqrt(1690000 - 2400 450 + 450^2) = 0:
So this solution is correct

x = 450 - D = sqrt [ 450^2 + 300^2 ] + sqrt [ (1200 - 450)^2 + 500^2 ]

Distance =~1,442 meters.

 Nov 6, 2018

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