2 ft/sec 20 ft 6 ft

Math 181

1)
A spherical balloon is being inflated in such a way that its radius is growing
at a rate of 2 cm/sec.

How
fast is the volume changing when the radius is 10 cm?

2)a)
A more realistic model is one in which the volume of the balloon is changing at
a constant rate (ie, air is being blown in to the
balloon at a constant rate). If air is
being blown in at a rate of 1000 cm^{3}/sec, how fast is the balloon
expanding when its radius is 4 cm?

b) How fast is the balloon expanding when the
radius is 20 cm?

c) How fast is the balloon expanding after 3
seconds of inflation?

(Hint: first figure out how
much air is in the balloon after 3 seconds, then figure out what the radius
is.)

*After 3 seconds, the volume
is 3000 cm*^{3}*/sec. Now repeat as in parts (a) and (b).*

3) When gravel is poured onto a pile, it forms a
right-circular cone where the height is always 3 times the radius. If gravel is poured at a rate of 2 ft^{ 3} / min , determine:

*(Note: a formula for the volume of a cone can be
found on the inside cover of your book)*

a)
What is the volume after 2
minutes of pouring?

4 ft^{ 3}

b) What is the radius after 2 minutes of
pouring?

** h **

**r** c) How fast is the radius
growing after 2 minutes?

4) A
plane is flying at an altitude of 5 miles, with a speed of 600 miles per hour.

a)
At the instant the plane
passes directly over your head, how far is the plane from you?

*5 miles*

b)
10 miles

How
far is the plane from you 1 minute later?

By the Pythagorean theorem,
it is

c)
What was the plane's __average__
speed relative to you during that minute? *(remember how to find average speeds)*

*Change in distance over
change in time: (11.18-5miles)/1min – 6.18
miles/min = 370.8 miles/hr.*

d)
What is the plane's
(instantaneous) speed relative you after one minute?

*Let h = the horizontal
distance of the plane from you, and z = the total distance. Then *

* *