Doing Related Rates with Maple

Suppose a point P  = ( x , y ) moves counter-clockwise along the circle of radius . Suppose in addition that when P = (1,3), . What is  at this point? This is a simple example of a related rate. The variables x  and y  are related by the equation . However both x  and y  are in fact functions of time t . We indicate the dependence on t  by writing

If we differentiate this equation with respect to t  (with the help of the chain rule), we obtain an equation relating  and . We let Maple perform these calculations. We begin by assigning the above equation a name.

>	eq1:=x(t)^2+y(t)^2=10;

We now use the diff  command to differentiate this equation with respect to t . We call the resulting equation eq2 .

>	eq2:=diff(eq1,t);

We now solve eq2  for  and call it

>	v[y]:=solve(eq2,diff(y(t),t));

Finally we substitute  and  if .

>	subs({x(t)=1,y(t)=3,diff(x(t),t)=-1.5},v[y]);

Thus we find  at this point.

The main ideas behind doing related rates with Maple include

• Express all variables as functions of t
• Use the diff command to differentiate with respect to t
• Use the solve and subs command effectively. In particular you can use a derivative just like any other variable.

Project

Let k  denote the number of letters in your first name. A light is loacted at the point (0,4). A small ball located at point P  = ( x , y ) moves counter-clockwise on the circle of radius 1 centered at (0,1). The light casts a shadow of the ball at point Q = (z,0) on the horizontal axis.

The following animation illustrates what is happening.

Suppose  when P  is at the point (   ).

1) Determine  at this instant.

2) Determine the rate at which the shadow Q  is moving on the horizontal axis at this instant.

Suggestions:

• Write down the equation of the circle and make x  and y  functions of t .
• Differentiate this equation with respect to t  to get an equation relating  and . Use this equation to calculate .
• Let ( x , y ) be a point on the circle and let ( z ,0) denote the corresponding point of the shadow on the x -axis. Express z   in terms of x  and y . Use either similar triangles or the equation of the line passing through the points  (0,4) and P . Warning:  If you use x  and y  as the variables in the equation of a line, then you must temporarily relabel the coordinates of P.
• Differentiate this equation to get an equation relating  to  and . Use this equation and the previous result to calculate  at this instant.

Extra Credit

Determine the coordinates of the ball on the circle when the shadow is at the farthest distance from the origin on the positive x -axis. Explain your solution.