The Beach Race

A beach race begins from a point, 4 km out to sea from one end of a 10km beach. Racers must swim to a point along the beach, and then run to the finish line.

If I can swim at 4 kph and run at 10 kph, to what point on the beach should I aim to land?

Can you find the landing point for which the time of the race will be as short as possible - as accurately as possible?

Assessment

Be sure you have activated CAS before you begin.

Hint: When entering mathematical expressions in the math boxes below, use the space key to step out of fractions, powers, etc. On Android, begin entry by pressing Enter.

Type simple mathematical expressions and equations as you would normally enter these: for example, "x^2[space]-4x+3", and "2/3[space]". For more interesting elements, use Latex notation (prefix commands such as "sqrt" and "nthroot3" with a backslash (\)): for example: "\sqrt(2)[space][space]". More?

1. Drag point C to find the landing position which will result in the LONGEST time for the race.

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2. There are two landing points which will result in a race time of exactly 2 hours. One of these is to swim directly to the point O and then run the full length of the beach (swimming the shortest possible distance). Drag the point C to find the other landing point which will result in a 2 hour race.

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3. Now build an algebraic model for the times for the race, letting the distance OC = x: swimtime(x) =

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4. runtime(x) =

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5. racetime(x) =

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6. Finally, plot the derivative of this RaceTime(x) function to locate the best place to land: myderiv(x) =

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7. myderiv(x) = 0 when x =

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How difficult did you find this activity? Please assign a rating value from 1 (very easy) to 5 (extremely difficult). Comments? Suggestions? What did you learn from this activity?

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This dynamic figure was created using the latest version of the free GXWeb from Saltire Software, as shown below.