Question 1097839
<br>Let A be the point where you are standing; let B be the point on the east bank directly opposite of where you are standing: let C be the point on the east bank where you will beach your canoe and start running; and let D be the point where your friends are waiting for you.<br>
You are looking for the minimum total time required to reach your friends.  That means you need to form an expression in terms of some variable for the total time and find the minimum value of that expression. Finding the minimum value means finding where the derivative of the total time expression is equal to zero.<br>
Let length BC be x; then CD is (10-x).<br>
AC is the hypotenuse of right triangle ABC with legs of length 3 and x; you will be rowing that distance at a rate of 4.5 km/h.  CD is the distance you will be running at 7 km/h.  The total time required is<br>
{{{sqrt(x^2+9)/4.5 + (10-x)/7}}}
or
{{{(x^2+9)^(1/2)/4.5 + (10-x)/7}}}<br>
The derivative of this expression is<br>
{{{(1/2)(x^2+9)^(-1/2)*(2x)/4.5 - 1/7}}}<br>
Setting this expression equal to 0 and solving for x...
{{{x/(4.5*sqrt(x^2+9)) = 1/7}}}
{{{7x = 4.5*sqrt(x^2+9)}}}
{{{49x^2 = 20.25(x^2+9)}}}
{{{49x^2 = 20.25x^2+182.25}}}
{{{28.75x^2 = 182.25}}}
{{{x^2 = 182.25/28.75}}}
{{{x = sqrt(182.25/28.75) = 2.517763}}}  approximately<br>
Substituting this value into the expression for the amount of time required (I won't show the arithmetic!) gives a minimum time of about 1.93923 hours.<br>
This result was verified by using a graphing calculator to find the minimum value of the expression for the total amount of time required.