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 cm3/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 cm3/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