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Problem setting: A man with a boat at point S at sea wants to get to point Q inland. Point S is D1=4[km] from the closest point P on the shore, point Q is D2=2[km] from the closest point T on the shore and point P and T are at a distance of D=8[km] .

 

Questions: If the man rows with a speed of v_r =3 [km/hr] and walks with a speed of v_w =7 [km/hr] at what point R should he beach the boat in order to get from point S to point Q in the least possible time? What is the distance and time rowed? What is the distance and time walked?

 

Suggested solution steps

Sketch the problem and setup the function to be minimized.

(a) Where is point R in comparison to the point on the shore where the line joining S and Q would cross? Make a sketch of the problem with x the distance between P and R.
(b) What variable needs to be minimized? Find a formula for this variable as a function of x .

Find the x -value for the minimum and substitute back.

(c) Using a graphing calculator and trace function find x at the minimum of this function. This is the value of R. (Or find the minimum be numeically finding the roots of the derivative.)

(d) Substitute back to find the distance and time rowed and the distance and time walked?

Solution steps

Sketch the problem and setup the function to be minimized.

(a1) Where is point R in comparison to the point on the shore where the line joining S and Q would cross?

If the man rowed and walked at the same speed, the quickest path would be the straight line joining S and Q. Assume this line crosses the shoreline at R'. Because the man can walk faster than he can row, we would expect that he will spend less time rowing than if he had traveled this line, so that he will land to the left of R'.

(a2) Sketch

(b1) What variable needs to be minimized?

The total time t

(b2) Find this variable as a function of x

We need to find t as a function of x .

t(x)=t_r(x)+t_w(x) where t_r(x) is the time rowing and t_w(x) is the time walking.

Find the distance rowing, that is the length of \overline {SR} and the distance walking, that is the length of \overline {RQ}

\overline {SR} = \sqrt {4^2 + x^2 } = \sqrt {x^2 + 16}

\overline {RQ} = \sqrt {(8-x)^2 + 2^2 } = \sqrt {x^2 -16x + 68}

Find the time for rowing and for walking.

t_r = {{\overline {SR} } \over {v_1 }} = {{\sqrt {x^2 + 16} } \over 3}

t_w = {{\overline {RQ} } \over {v_2 }} = {{\sqrt {x^2 -16x + 68} } \over 7}

Find the total time.

t=t_r+t_w = {{\sqrt {x^2 + 16} } \over 3} + {{\sqrt {x^2 -16x + 68} } \over 7}

 

Find the x -value for the minimum and substitute back.

(c1): Using a graphing calculator and trace function find x at the minimum of this function. This is the value of R.

               

(c2) Check with Newton's method?

Find the minimum of t(x) by finding roots of \frac{\text{d}t}{\text{d}x} using Newton's method.

\frac{{{\rm{d}}t}}{{{\rm{d}}x}} = \frac{x}{{3\sqrt {x^2 + 16} }} + \frac{{x - 8}}{{7\sqrt {x^2 - 16x + 68} }}

 

For Newton's method we need the derivative of \frac{\text{d}t}{\text{d}x} , that is the second derivative.

\frac{{{\rm{d}}^2 t}}{{{\rm{d}}x^2 }} = \frac{4}{{3 \cdot \left( {x^2 + 16} \right)^{3/2} }} + \frac{{16}}{{7 \cdot \left( {x^2 - 16x + 68} \right)^{3/2} }}

 

n x_n           f(x_n)         f'(x_n)     x_n-f(x_n)/f'(x_n)
0 0.00000000 -0.13859179 0.08435239 1.64300958
1 1.64300958 -0.00962233 0.06788675 1.78475052
2 1.78475052 -0.00016733 0.06551957 1.78730444
3 1.78730444 -0.00000005 0.06547662 1.78730528
4 1.78730528 0.00000000 0.06547661 1.78730528
5 1,78730528      

The result is the same: R=1.8

 

(d) Substitute back to find the distance and time rowed and the distance and time walked?

d_r = \overline {SR} = \sqrt {x^2 + 16} = \sqrt {1.79^2 + 16} = 4.38

t_r = {{d_r } \over {v_r }} = {{4.38} \over 3} = 1.46
d_r = \overline {RQ} = \sqrt {x^2 -16x + 68} = \sqrt {1.79^2 -16x + 68} = 6.53
t_r = {{d_w } \over {v_w }} = {{4.38} \over 3} = 0.93
t_{Total}=t_r+t_w=2.93

 

Solution:

The point R is 1.8 km from point P.

 

d_r = 4.38    t_r = 1.46   

 

d_r = 6.53    t_r = 0.93

 

t_{Total}=t_r+t_w=2.93

 

A GeoGebra interactivity showing concerns about various numerical methods is given here.

 


Page last modified on November 07, 2007, at 09:17 AM