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Read the problem description.
Problem setting: A man with a boat at point S at sea wants to get to point Q inland. Point S is distance d1 from the closest point P on the shore, point Q is distance d2 from the closest point T on the shore. The points P and T are at a distance of d from each other.

 

 

Question: If the man rows with a speed of v_r and walks with a speed of v_w at what point R should he beach the boat in order to get from point S to point Q in the least possible time?

 

Watch animated demo of the simulator.

               

Start the simulator.

Questions

1. Let's check the simulator!

If you made your own simulator - use these questions to make sure it is working properly!
A. Set the parameters d=4, d1=3 and d2=0.  

  • Notice that d2=0 moves the point Q onto the shoreline (Q=T).
    • What is the distance from S to Q?
 
  • Now move the slider point R=4 (the length of d).
    • Describe the way the man gets from S to Q?
 
  • Slide v_r =1.
    • How long does it take him to row directly from S to Q? Do you see Time=5 ? Why?
 
If you do not see Time=5 , it may be that you have an error in one of your functions for rowing.
  • Slide v_r =2 .
    • Do you see Time=2.5 ? What does this mean? What do you expect Time to be if v_r =5 ?  
 
  • Why doesn't it matter what v_w is?

B. Set the parameter d=7. You should have d1=3, d2=0 and v_r =2 .

  • Slide v_w =6 and move the slider point R=4.
    • Describe the way the man gets from S to Q?   
 
  • Do you see Time=3 ? Why?
 

If you do not see Time=3 , it may be that you have an error in one of your functions for walking.
C. Set the parameters d=8, d1=3 and d2=3, v_r =5 and v_r =10.

  • Now move the slider point R=4 (half the length of d).
    • Describe the way the man gets from S to Q?   
 
  • How long did it take him to row from S to R?
 
  • How long did it take him to walk from R to Q?
 
  • How long does it take him to get from S to Q via R? Do you see Time=1.5 Why?
 

 

2. Let's understand the problem!

Use these questions to make sure you understand the problem.
A. Check that your parameters are d=8, d1=3 and d2=3, v_r =5 and v_r =10.

  • Check that the slider point R=4 (half the length of d).
    • Is this path from S to Q via R the shortest distance between S and Q? Why?   
 
  • Do you think that this is the fastest route for him to travel?
  • Check that the Time=1.5 and remember why.
 
  
  • Move the slider point R=2. Check that the Time=1.39 . Is this more or less than 1.5?
    • Is this a faster route for him to travel than along the diagonal?   
 
  • So - for these parameters - is the fastest route, the shortest route?   
 

B. Try to describe why this is so. You can use both numbers and words.   

He rows \overline{SR}=\sqrt{3^2+2^2}=3.6km . At v_r =5km/hr, t_r=\frac{3.6}{5}=0.72hr   

He walks \overline{SR}=\sqrt{3^2+(8-2)^2}=\sqrt{3^2+(6)^2}=6.7km . At v_r =10km/hr, t_r=\frac{6.7}{10}=0.67hr   
Total time is: t=t_w+t_r=0.72+0.67=1.39hr

 

He rows much slower than he walks - in fact, twice as slow. So he probably should spend more of his trip walking than rowing. (This last sentence is actually difficult to write! Should he spend more "time"?  ... more "distance"?  ... more "of his trip"?)

 

C. Above we had rowing speed less than walking speed. Let's change this relationship.

  • Now let v_r =10 and v_r =5.
    • Did the picture change? Did you see the green Time function t shift? If not change the sliders again. When you get to the questions 4, remember this!
      • How long did it take him to row from S to R?
 
  • How long did it take him to walk from R to Q?
 

 

3. Let's try the simulator! - Record and check your values online.

  1. Set d=7, d1=4, d2=2, v_r =3 and v_w =7 and then slide R=1.75. Is this the point at which the Time t is minumum?
    • Record in row 1 these values of R and T in the table below.
  2. Using the simulator, set the parameters: d=8, d1=3, d2=4, v_r =7 , v_w =2 and then find the minimum Time t and the point R.
    • Record in row 2 these values of R and T in the table below.
  3. Choose a couple of sets of parameter values and then find the minimum Time t and the point R.
    • Record in rows 3 and 4 the parameter values and the values of R and T in the table below.

 

4. Let D be the point where the diagonal joining S and Q passes through the shoreline \overline{PT} .

  • Draw the diagonal \overline{SQ} : Click on the segment icon . Click on the point S and then on the point Q. The line segment a should appear.
  • Draw the point D: Click on the intersect icon . Click on the new line a and then on the x-axis. Select any of the choices. The point D should appear.

 

  1. If the man rows slower than he walks, where should the point R be in relation to the point D?
    • Make up some parameters and test the simulator.
  2. If the point R is to the right of D what can you say about the relationship between v_r and v_w ?
    • Make up some parameters and test the simulator.

 

5. What happens if the speed rowing equals the speed walking?

  1. Now just change v_r=2 and v_w =2. Use the simulator to find the minimum Time t and the point R.
    • Record in row 6 these values of R and t in the table below. Look carefully at the route the man travels. Does it look straight?
  2. Change v_r=3 and v_w =3. Use the simulator to find the minimum Time t and the point R.
    • Record in row 7 these values of R and t in the table below. Are they the same as in question 1?  Again look carefully at the route the man travels.
  3. What do you think R and t will be for v_r =4= v_w , v_r =5= v_w ? Check your answer using the simulator.
    • Record in rows 8 and 9 these values of R and t in the table below. What is the route that the man travels? Discuss why you think this line is the straight line joining S and Q?
  4. Change the parameters d, d1 and d1 and then make v_r = v_w (use different values – say v_r =2=v_w , v_r =3=v_w , …) What is the route that the man travels?
    • Can you draw a conclusion.  That is: If the man rows at the same speed that he walks, what route should he travel from S to Q?
  5. Fix d, d1 and d2 (for example d=8, d1=3, d2=4). Make a scale drawing on paper and then draw the diagonal from S to Q.  
    Can you find the point R using just 5th grade geometry – that is, without using the simulator?
    • How would you check your answer using the simulator?
           Set the speeds equal to each other (for example v_r =3=v_w ) and find the minimum Time t and the point R.

 

InterActive Spreadsheet
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When you are done - scroll to the right to check your answers!

Requires: SunJava Player for Simulator and Demo Flash Player for Demo
Author: Linda Fahlberg-Stojanovska Contact for offline use.
Printing: Printing this page should open all of the toggles - except the answers!
Other: Eventually  
  1. Good Questions for the Boat-Landing Problem: AP Calculus Problem (Extreme Value)
  2. Good Questions for the Boat-Landing Problem: Numerical Solutions Problem

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Page last modified on April 06, 2010, at 12:18 PM