Question 1162238
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Let x be the distance from P where the man reaches shore.  Then the distance from there to the village is 12-x.<br>
The distance the man rows, by the Pythagorean Theorem, is {{{sqrt(x^2+9)}}}.<br>
The time the man takes to row to the shore is {{{sqrt(x^2+9)/2.5}}}<br>
The time he takes to walk from there to town is {{{(12-x)/6.5}}}<br>
We want to find x that makes the total time a minimum:<br>
{{{y = sqrt(x^2+9)/2.5 + (12-x)/6.5}}}<br>
{{{dy/dt = (0.5(2x))/(2.5sqrt(x^2+9))-1/6.5}}}<br>
Set the derivative equal to 0 and solve for t:<br>
{{{(0.5(2x))/(2.5sqrt(x^2+9)) = 1/6.5}}}<br>
{{{x/(2.5sqrt(x^2+9)) = 1/6.5}}}<br>
{{{6.5x = (2.5sqrt(x^2+9))}}}<br>
{{{(13/5)x = sqrt(x^2+9)}}}<br>
{{{(169/25)x^2 = x^2+9}}}<br>
{{{(144/25)x^2 = 9}}}<br>
{{{x^2 = 225/144}}}<br>
{{{x = 15/12 = 5/4 = 1.25}}}<br>
ANSWER: The man should land the boat 1.25 miles from P.<br>
The answer can be confirmed by finding the minimum value of the time y using a graphing calculator.<br>