Name
Date
Exam #3 - Related Rates
Calc
Answer all the questions in an orderly fashion. Please clearly indicate your answer and
:e sure to labe
estionT
1) A fire has started in a dry, open field and spreads in the form of a circle. The radius of
the circle increases at a rate of 6 ft/min. Find the rate at which the area is increasing
when the radius is 150 ft.
2) A water baUoon is filled at a rate of 20 cubic inches per second. Assume the balloon is
a perfect sphefl. How fast is the radius of the balloon increasing when the radius is 10?
p ,50
r dt:
3)
10 cm
—•)
A container has rhe shape of an open right circular cone, as shown in the figure above. The height of the
container is 10 cm and the diameter of the opening is 10 cm. Water in the container is evaporating so that
its depth h is changing at the constant rate of -^ cm/hr.
(Note: The volume of a cone of height h and radius r is given by V = -Lrr2A.)
(a) Rod the volume V of water in (he container when /i = 5 cm. Indicate units of measure,
(b) Find the rate of change of the volume of water in the container, with respect to time, when ft = 5 cm.
Indicate units of measure.
V-
lo
K
/O
dt:
jy -^TTT riStif <-***/
/*<
it ^T V
Ship A is traveling due west toward Lighthouse Rock at a speed of 15 kilometers per hour (km/hr). Ship B is
traveling due north away from Lighthouse Rock at a speed of 10 km/hr. Let x be the distance between Ship A
and Lighthouse Rock at time /, and let y be the distance between Ship B and Lighthouse Rock at time t, as
shown in the figure above.
(a) Find the distance, in kilometers, between Ship A and Ship B when jr = 4 tain and y = 3 km
/
3
Cl TjV
/ \ fr\"' r
<W Find the rate °1 Change, in km/hr, of the distance between the two ships when or = 4 km and y = 5 km,
(c) Let 0 be the angle shown in the figure. Find the rate of change of S. in radians per hour, when jt = 4 km
and v = 3 km.
£ —z
oc
^ /-
-*(&*
5) A street light is mounted at the top of a 15 foot pole. A man 6 feet tall walks away
from the pole with a speed of 5 feet/sec along a straight path.
a) How fast is the tip of his shadow moving when he is 40 feet from the pole?
b) How fast is the length of his shadow changing?
c*Vr
rr
A coffeepot has the shape of a. cylinder with radius 5 inches, as shown in the figure above. Let A be the depth of
the coffee in the pot, measured in inches, where A is a function of time /. measured in seconds. The volume V
of coffee in the pot is changing at the rate of -5W7T cubic inches per second, (The volume V of a cylinder with
radius r and height h is V = x^h.)
(a)
Bonus:
Oil is leaking from a pipeline on the surface of a lake and forms an oil slick whose volume increases at a
constant rate of2QOO_£uj>ic centimeters per minute. The oi! slick takes the form of a right circular cylinder
with both its radius and height changing with tune. (Note: The volume V of a right circular cylinder with
radius r and height h is given by V = >rr2h, )
the oil slkk with respect to time, in centimeters per minute?
AV