Best Answer bonanova, 29 March 2013 - 09:33 AM

Spoiler for The time depends on

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The distance x from point A that the lifeguard enters the water.

Following the path MAC, (x=0) his time to the swimmer is 58.08 seconds

Following the path MBC, (x=280) his time to the swimmer is 87.80 seconds.

Following the straight path MC, (x=112) his time to the swimmer is 60.22 seconds.

So there is a value of x, less than 112, that minimizes his time to the swimmer.

Snell's law says the path of least time occurs at interfaces of differing speeds

when the sines of the angles with the normal are in the same ratio as the speeds.

We can have some fun deriving that result here.

The distance run on sand increases with x as sin(a) where a is his angle to the perpendicular to the shore.

The distance swum in the water decreases with x as cos(b) where b is his angle with the shore.

Since the speed ratio is 2, the stationary condition is 2 sin (a) = cos (b).

If we had used angle with normal in the water, we would have 2 sin (a) = sin (b) which is Snell's law.

This occurs when

Thus tan (a) = 0.5 and we get

The sand distance is 89.44 meters (22.36 seconds) and water is 268.33 m (33.54 seconds.)

So the minimum time to the swimmer of

Following the path MAC, (x=0) his time to the swimmer is 58.08 seconds

Following the path MBC, (x=280) his time to the swimmer is 87.80 seconds.

Following the straight path MC, (x=112) his time to the swimmer is 60.22 seconds.

So there is a value of x, less than 112, that minimizes his time to the swimmer.

Snell's law says the path of least time occurs at interfaces of differing speeds

when the sines of the angles with the normal are in the same ratio as the speeds.

We can have some fun deriving that result here.

The distance run on sand increases with x as sin(a) where a is his angle to the perpendicular to the shore.

The distance swum in the water decreases with x as cos(b) where b is his angle with the shore.

Since the speed ratio is 2, the stationary condition is 2 sin (a) = cos (b).

If we had used angle with normal in the water, we would have 2 sin (a) = sin (b) which is Snell's law.

This occurs when

**x = 40**, and coincidentally when a = b.Thus tan (a) = 0.5 and we get

**a = 26.56**^{o}.The sand distance is 89.44 meters (22.36 seconds) and water is 268.33 m (33.54 seconds.)

So the minimum time to the swimmer of

**55.90 seconds**is achieved by entering the water 40 meters from point A.