Worksheet on Related Rates

Recall the Chain Rule:

If z depends on x, and x depends on t, then we have This can also be written as or, if you prefer, as  Let’s apply this in the case that z = x3: Let’s try the case that z = sin x. Now, find each of the following:
 
 

    2.  

    2.  

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    4.  
      5.  
  1. Suppose a rectangle has a base that is increasing at a rate of 4 inches per minute and a height that is increasing at a rate of 7 inches per minute.
    1. Sketch a picture of the rectangle below with variables included:

      2.  
    2. Translate the rates given above into expressions involving derivatives.

      3.  
    3. Write an expression for the area of the rectangle in terms of the variables used in part a.

      4.  
    4. Differentiate each side of the expression with respect to t. (i.e., apply  to both sides.)

      5.  
    5. How fast is the area of the rectangle increasing when the length of the base is 15 inches and the height is 25 inches?

 

The area is increasing at a rate of 205 in2/minute.
     
     
  1. A person parachutes out of a plane. Due to a strong wind, the parachutist falls straight down. After the parachute opens, the parachutist begins falling at a constant rate of 14 miles per hour. The plane continues flying at a speed of 100 miles per hour. How fast is the distance between the parachutist and the plane increasing after the parachutist has fallen 0.2 miles and the plane has flown 1 mile past the drop site?



    2.  
     

       
    The distance between the parachutist and the plane is increasing at a rate of 100.804 miles per hour.
     
     
  2. Before releasing a bird into the wild, two scientists equipped it with a device that is able to measure the distance between themselves and the bird as well as the rate at which that distance is changing. They now (several months later) hope to catch this bird in order to examine it and make sure it is thriving. They can see that the bird is heading due south towards a clearing in the woods (where it will be easy to catch) and is 400 meters away from that clearing. The scientists are 300 meters due west of the clearing. If the distance between them and the bird is decreasing at a rate of 3.6 meters per second and they are heading toward the clearing at a speed of 2 meters per second, how fast is the bird travelling? Will they reach the clearing before the bird?

 
Since the triangle is a 3-4-5 triangle, we can see that c = 500, or we can use the Pythagorean Theorem:     The bird is flying at a speed of 3 meters per second toward the clearing. The bird will reach the clearing in 400/3 = 133.33 seconds. They will reach the clearing in 300/2 = 150.00 seconds. They will not be able to reach the clearing before the bird.