1. Related Rates

2. Example 1: If the radius of a circle is
increasing at the rate of 7cm/sec, how
fast is its area increasing when the radius
is 20 cm? r r r

3. Solution to Example 1:
Let A be the area and r the radius of
circle at time t.
Know
Find

4. By the chain rule,

5. Example 2: One end of a 13-foot ladder
is on the ground and the other end rests
on a vertical wall. If the bottom end slides
away from the wall at the rate of 3 ft/sec,
how fast is the top of the ladder sliding
down the wall when the bottom of the
ladder is 5 feet from the wall?

6. Differentiating implicitly w.r.t. x, we get

8. Example 3: An object is moving
clockwise around the circle
As it passes through the point
the y-coordinate is decreasing
at the rate of 3 units per second. At
what rate is its x-coordinate changing
at that point?

9. Solution to Example 3:
Know
Find

11. Example 4: A conical paper cup 8 cm
across the top and 12 cm deep is full of
water. Water begins to leak out of the
bottom at the
rate of
How
fast is the level of water dropping when
the water is 3 cm deep?

12. Let V, r, and h denote the volume, radius,
Solution to Example 4:
Let V, r, and h denote the volume, radius,
and hight, respectively, of the water
remaining at time t
Know
Find

15. Example 5: The area of an equilateral
triangle is increasing at a rate of
Find the rate the length
of each side is increasing when the
area is

16. Solution to Example 5:
Let A, x, and h denote the area, length of
each side, and the height, respectively, of
the triangle at time t min.
Know
Find