2.6 Related

RateProblems

Def: Related Rates are the rates of change of two or more related variables that are changing with respect to time.

The basic idea behind related rates is that each of the variables is now a function of time, or t.

When we differentiate, we will do so with respect to t.

Some of the variables will not be changing with respect to t. These variables are like constants, and their derivatives with respect to t is 0.

Related Rates

Solving Related Rates

Step 1: Identify the changing quantities, possibly with the aid of a sketch.

Step 2: Write down an equation that relates the changing quantities.

Step 3: Differentiate both sides of the equation with respect to t.

Step 4: Solve for the required quantity, and substitute the given values in the derived equation.

Example 1: One day Ms. Henry arrived home late from work because she was tutoring

calculus students. Her geriatric dog, Kelty, managed to hold herself for 13 hours, but when Ms. Henry let her out, and she noticed the area of Kelty’s circular doggie puddle was growing at a rate of 12 square inches per second.

a) How fast was the radius growing at the instant when it equaled 20 inches?

b) How fast was the circumference changing at the same time?

Example 2:

Ms. Henry accidentally hit a calculus student in the head with a snowball. In retaliation, the student formed a snowball to return the fire, and as it was being formed, it was melting. If the snowball is melting at the rate of 10 cubic inches per minute, at what rate is the radius changing when the snowball is 2 inches in radius?

http://www.mathopenref.com/calcsnowballproblem.html

Example 3: A 25 foot ladder is leaned against the

wall of a building. Because it is not placed on stable ground, the base of the ladder begins slipping away from the wall at a rate of 2 feet per second.

a) How fast is the top moving down the wall when the base of the ladder is 7 feet from the wall?

b) Consider the triangle formed by the side of the house, the ladder, and the ground. Find the rate at which the area of the triangle is changing when the base of the ladder is 7 feet from the wall.

http://astro.ocis.temple.edu/~dhill001/relatedrates/ladderdemo.gif

Example 4: Dry sand is loaded onto a conveyor belt. The conveyor belt moves sand up at a fixed rate, and it spills off to form a conical pile, with the height equal to the radius. How fast is the height increasing when the height is 3 feet and sand is being added at a rate of 2 cubic feet per minute?

http://astro.ocis.temple.edu/~dhill001/relatedrates/sandpiledemo.mov