1. Related Rates
Section 4.6a

2. Do Now
Write the formula for distance from the origin to a point… and find its derivative!
Suppose that the coordinates of the point (x and y) are both
differentiable with respect to t. Then we can use the Chain
Rule to find an equation that relates dD/dt, dx/dt, and dy/dt:
Any equation involving two or more variables that are
differentiable functions of time t can be used to find an
equation that relates their corresponding rates!!!

3. Related Rates Problems
Examples:
How fast is a balloon rising at a given instant?
How fast does the water level drop when a tank
is drained at a certain rate?
How fast does the surface area of a bubble
increase when its volume increases at a certain
rate?
Questions like these require us to calculate a
rate that may be difficult to directly measure,
using a rate that we know, or is easy to
measure  RELATED RATES!!!

4. Related Rates Problem Strategy
1. Draw a picture and name the variables and
constants. Use t for time. Assume all variables
are differentiable functions of t.
2. Write down the numerical information (in terms
of the symbols you have chosen).
3. Write down what we are asked to find (usually a
rate, expressed as a derivative).

5. Related Rates Problem Strategy
4. Write an equation that relates the variables. You
may have to combine two or more equations to get
a single equation that relates the variable whose
rate you want to the variables whose rates you
know.
5. Differentiate with respect to t. Then express the
rate you want in terms of the rate and variables
whose values you know.
6. Evaluate. Use known values to find the unknown
rate.

6. Guided Practice
When a circular plate of metal is heated in an oven, its
radius increases at the rate of 0.01 cm/sec. At what rate
is the plate’s area increasing when the radius is 50 cm?
Step 1: r = radius of plate, A = area of plate
Step 2: At the instant in question, dr/dt = 0.01 cm/sec,
r = 50 cm
Step 3: Need dA/dt
Step 4:

7. Guided Practice
When a circular plate of metal is heated in an oven, its
radius increases at the rate of 0.01 cm/sec. At what rate
is the plate’s area increasing when the radius is 50 cm?
Step 5:
Step 6:
At the instant in question, the area is increasing at
the rate of

8. Guided Practice
A hot-air balloon rising straight up from a level field is tracked by
a range finder 500 ft from the lift-off point. At the moment the
range finder’s elevation angle is , the angle is increasing
at the rate of 0.14 rad/min. How fast is the balloon rising at
that moment?
rad/min
when y
when
Range
Finder
500 ft

9. Guided Practice
A hot-air balloon rising straight up from a level field is tracked by
a range finder 500 ft from the lift-off point. At the moment the
range finder’s elevation angle is , the angle is increasing
at the rate of 0.14 rad/min. How fast is the balloon rising at
that moment? y
At the moment in question, the balloon
is rising at the rate of 140 ft/min
500 ft

10. Guided Practice
A police cruiser, approaching a right-angled intersection from
the north, is chasing a speeding car that has turned the corner
and is now moving straight east. When the cruiser is 0.6 mi
north of the intersection and the car is 0.8 mi to the east, the
police determine with radar that the distance between them and
the car is increasing at 20mph. If the cruiser is moving at 60mph
at the instant of measurement, what is the speed of the car? y
Situation when x = 0.8, y = 0.6 s y x x

11. Guided Practice y x Situation when x = 0.8, y = 0.6 s y xx
Pythagorean Theorem:

12. Guided Practice y x Situation when x = 0.8, y = 0.6 s y xx
At the moment in question,
the car’s speed is 70 mph

13. Guided Practice
Water runs into a conical tank at the rate of The tank
stands point down and has a height of 10 ft and a base radius
of 5 ft. How fast is the water level rising when the water is
6 ft deep?
5 ft x
10 ft y
when y = 6 ft

14. Guided Practice
Water runs into a conical tank at the rate of The tank
stands point down and has a height of 10 ft and a base radius
of 5 ft. How fast is the water level rising when the water is
6 ft deep?
5 ft x
Similar Triangles:
10 ft y

15. Guided Practice
Water runs into a conical tank at the rate of The tank
stands point down and has a height of 10 ft and a base radius
of 5 ft. How fast is the water level rising when the water is
6 ft deep?
5 ft x
10 ft y
At this moment, the water level is rising at about ft/min.