We use cookies to personalise content and advertisements and to analyse access to our website. Furthermore, our partners for online advertising receive pseudonymised information about your use of our website.
Please click on "Accept cookies" if you agree to the setting of cookies. Cookies that do not require consent remain unaffected by this, see
cookie policy and privacy policy.
DECLINE COOKIES

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.

Mathgenius Nov 6, 2018

#1**0 **

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

Mathgenius Nov 6, 2018

#2**0 **

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

dierdurst Nov 6, 2018

#3**0 **

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.**

Guest Nov 6, 2018