1. Related Rates
Section 4.6b

2. Do Now: #10, p.238
Suppose that the edge lengths x, y, and z of a closed
rectangular box are changing at the following rates:
m/sec
m/sec
m/sec
Find the rates at which the box’s (a) volume, (b) surface
area, and (c) diagonal length are changing at the
instant when x = 4, y = 3, and z = 2.

3. Do Now: #10, p.238
m/sec
m/sec
m/sec
x = 4 m, y = 3 m, z = 2 m
(a)

4. Do Now: #10, p.238
m/sec
m/sec
m/sec
x = 4 m, y = 3 m, z = 2 m
(b)

5. Do Now: #10, p.238
m/sec
m/sec
m/sec
x = 4 m, y = 3 m, z = 2 m
(c)

6. More Practice Problems
A spherical balloon is inflated at the rate of
How fast is the balloon’s radius increasing at the
instant the radius is 5 ft?
At the instant in question:

7. More Practice Problems
A spherical balloon is inflated at the rate of
(b) How fast is the surface area increasing at the same
instant?

8. More Practice Problems
A particle P(x, y) is moving in the coordinate plane in such a way
that dx/dt = –1 m/sec and dy/dt = –5 m/sec. How fast is the
particle’s distance from the origin changing as it passes through
the point (5, 12)?
P(5, 12) D
Need y
Based on the given information,
should dD/dt be positive or negative? x

9. More Practice Problems
A particle P(x, y) is moving in the coordinate plane in such a way
that dx/dt = –1 m/sec and dy/dt = –5 m/sec. How fast is the
particle’s distance from the origin changing as it passes through
the point (5, 12)?
P(5, 12) D y x
At the moment in question, the particle is
moving toward the origin at a rate of 5 m/sec

10. More Practice Problems
Now let’s work through #30 on p.240 together…
s = distance ball has fallen
x = distance from bottom of pole to
shadow
Light
Ball at t = 0
1/2 sec later
50-ft
pole
Need
Shadow x 30
x(t)

11. More Practice Problems
Now let’s work through #30 on p.240 together…
Use similar triangles to relate s and x.
Light
Ball at t = 0
1/2 sec later
50-ft
pole
Shadow x 30
x(t)

12. More Practice Problems
Now let’s work through #30 on p.240 together…
Differentiate with respect to t.
Light
Ball at t = 0
1/2 sec later
The shadow is
moving at a velocity
of –1500 ft/sec.
50-ft
pole
Shadow x 30
x(t)