Simon suffers an injury at a campsite on the bank of a canal

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SUMMARY

To determine the optimal landing point for Simon to minimize his time to the hospital, we analyze a right triangle formed by the canal's width and the distance he travels along the opposite bank. The canal is 6√2 km wide, and Simon paddles at 5 km/h and cycles at 15 km/h. The total time function, t = t_p + t_c, where t_p is the time spent paddling and t_c is the time spent cycling, must be minimized. The hypotenuse of the triangle is represented as c = 6√2 + x, where x is the distance traveled along the bank.

PREREQUISITES
  • Understanding of the Pythagorean theorem
  • Basic calculus, specifically differentiation
  • Knowledge of time-distance-speed relationships
  • Ability to interpret and analyze mathematical functions
NEXT STEPS
  • Learn how to apply the Pythagorean theorem in real-world scenarios
  • Study optimization techniques in calculus, focusing on finding maxima and minima
  • Explore time-distance-speed problems in physics
  • Practice solving similar problems involving multiple modes of transportation
USEFUL FOR

Students studying calculus, particularly those focusing on optimization problems, as well as educators looking for practical applications of mathematical concepts in real-life scenarios.

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Homework Statement


Simon suffers an injury at a campsite on the bank of a canal that is 6sqrt(2) km wide. if he is able to paddle his kayak at 5km/h and pedal a bicycle at 15km/h along the opposite bank, where should be land on the opposite bank to reach the hospital in minimum time?

Homework Equations


assuming Pythagorean theorem , etc.
when you find the equation, then find the derivative and make it = 0 to find max/min
answer

The Attempt at a Solution


I get the hypotenuse to c=6sqrt(2)+x but don't know where to go from here. Help would be appreciated. Diagram included and is an exact copy of the diagram that was provided with the question.
 

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ilii said:

Homework Statement


Simon suffers an injury at a campsite on the bank of a canal that is 6sqrt(2) km wide. if he is able to paddle his kayak at 5km/h and pedal a bicycle at 15km/h along the opposite bank, where should be land on the opposite bank to reach the hospital in minimum time?

Homework Equations


assuming Pythagorean theorem , etc.
when you find the equation, then find the derivative and make it = 0 to find max/min
answer

The Attempt at a Solution


I get the hypotenuse to c=6sqrt(2)+x but don't know where to go from here.
What makes you think that is the hypotenuse? What you have is just the sum of the width of the canal and the distance down the other side. In your relevant equations, you list the theorem of Pythagoras. Do you know how to use it to find the hypotenuse of a right triangle?
ilii said:
Help would be appreciated. Diagram included and is an exact copy of the diagram that was provided with the question.
 
Simon paddles the hypoteneuse at 5km/h, taking time ##t_p##, then cycles the remaining distance at 15km/h, taking time ##t_c##. Total time ##t = t_p+t_c##, which will be a function of x.
 
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