# Calculus I - Related Rates

**Related Rates Problems**

Another common application of the derivative involved

situations in which two or more related quantities are changing

with time.

These are called **related rates problems.**

**Example
**Suppose the radius of a circle is increasing at 7 cm/s. How fast

is the area increasing when the radius is 20 cm?

**Guidelines**

1 If possible, draw a picture to illustrate the problem
and

label the pertinent quantities.

2 Set up an equation relating all of the relevant quantities.

3 Differentiate (implicitly) both sides of the equation
with

respect to time (t).

4 Substitute in the values for all known quantities and rates.

5 Solve for the remaining unknown rate.

**Example
**A painter is painting a house using a ladder 15 feet long. A dog

runs by the ladder dragging a leash that catches the bottom of

the ladder and drags it directly away from the house at a rate of

22 feet per second. How fast is the top of the ladder moving

down the wall when the top of the ladder is 5 feet from the

ground?

**Example
**For the previous situation, how fast is the angle between the

ground and the ladder changing when the top of the ladder is 5

feet from the ground?

**Example
**Two ships sail from the same port. The first ship leaves port at

1:00AM and travels east at a speed of 15 nautical miles per

hour. The second ship leaves port at 2:00AM and travels north

at a speed of 10 nautical miles per hour. Determine the rate at

which the ships are separating at 3:00AM.

**Example
**A revolving beacon in a lighthouse located 3 miles from a

straight shoreline makes 2 revolutions per minute. Find the

speed of the spot of light along the shore when the spot is 2

miles from the point on the shore nearest the lighthouse.

**Example
**Flour sifted onto waxed paper forms a pile in the shape of a

cone with equal radius and height. The volume of flour in the

pile is increasing at a rate of 7.26 in

^{3}/s. How fast is the height of

the flour increasing when the volume is 29 in

^{3}?

**Homework
**Read Section 3.8

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