Question 1137355
.


            While I agree with the  @mananth's  answers to  (a)  and  (b),  I totally disagree with his answer to  (c),  which is incorrect.



<pre>
Let x be the "x-coordinate" of the intermediate "target" point D between points C and B on the opposite bank of the river.


Then the total time t(x) to get from  A to B  is


    t(x) = {{{sqrt(3^2 + x^2)/6}}} + {{{(8-x)/8}}}  hours.     (1)


The plot of this function is shown in the figure below, and it clearly shows that the function t(x) has a minimum between 0 and 8.


{{{graph( 330, 330, -1, 8, -1, 3,
          sqrt(3^2 + x^2)/6 + (8-x)/8
)}}}

Plot y = {{{sqrt(3^2 + x^2)/6}}} + {{{(8-x)/8}}}


To find the value of x which provides the minimum to t(x), take the derivative of t(x)


    t'(x) = {{{(2x)/(2*6*sqrt(9 + x^2))}}} - {{{1/8}}},


equate it to zero and solve the obtained equation for x


    {{{x/(6*sqrt(9+x^2))}}} - {{{1/8}}} = 0,

    8x = {{{6*sqrt(9+x^2)}}}

    64x^2 = 36(9+x^2)

    64x^2 = 36*9 + 36x^2

    64x^2 - 36x^2 = 324

    x^2 = {{{324/28}}} = 11.571

    x = {{{sqrt(11.571)}}} = 3.4 (approximately).



<U>Answer</U>.  The target point to minimize time is 3.4 kilometers from C to B,


         giving time  t(3.4) = {{{sqrt(3^2 + 3.4^2)/6}}} + {{{(8-3.4)/8}}} = 1.331 hours.
</pre>

Solved.