Answers - Related Rates

1) In the baseball diamond, we have a man running from second to third. In order to rearrange this scenario into familiar terms, I will lean the diamond a bit to rest it on its lower left side, making the line the player is  running on the y-axis, and the "third to home" line is the x-axis. 
Step 1 - The variables:
y = 60
y' = -28 (It is decreasing by 28 every second, as he runs DOWNWARD)
x = 80 (This is a constant; when you differentiate, it will go to zero)
z = ??   (z is now the distance from him to the plate)
z' = ??
Step 2 - The equation relating our variables is x2 + y2 = z2  Now solve for z using this last equation with x = 80 and y = 60. I get z = 100.
Step 3 - Differentiate x2 + y2 = z2
0 + 2yy' = 2zz'    (x is a CONSTANT, stop bothering me about that everybody. The distance from third to home remains at 80 feet throughout the entire baseball game)
yy' = zz'   (I divide both sides by 2)
Note that z', or the changing distance, is what we have been asked to solve. 

Step 4 - Fill in the variables you know:  (60)(-28) = (100)(z')
Solve for z'. I'm not going to do everything for you. Incidentally, you will get a negative number for z. The reason is because when the player is running downward, y is decreasing, and therefore z is decreasing. The change in z is then going to be negative. However, when I phrased my question, I didn't ask how fast z is changing, I asked how fast it is decreasing, and it is decreasing at a positive rate. For example, if I run forward at 10 feet per second, then I run backward at -10 fps.

2) Step 1 -
x = 4
x2 = 3y2 + 7     42 = 3y2 + 7     y = √3
x' = √12
Step 2 -   x2 = 3y2 + 7
Step 3 -   2xx' = 3(2)yy'
Step 4 -   2(4)(√12) = 6(√3)(y')
y' = 8/3   (or dy/dt = 8/3)


Copyright 2001 Bruce