Problem :
Suppose a dog named Tika is chasing a duck in a straight line. If the duck's speed is
given by
d'(t) = 5
feet per second and Tika's speed by
T'(t) = 2t
feet per second, how
far has Tika traveled when her speed is equal to the duck's speed? If the duck gets a
100
foot head start, how far has Tika traveled when she catches the duck?
Figure %: The Dog Tika Chasing a Duck
Tika's speed is equal the duck's speed after
5/2
seconds. To compute the distance
she has traveled in this time, we integrate her speed from
0
to
5/2
:
2tdt = (t
^{2}_{0}
^{5/2}) =


To find how far Tika must run to catch the duck, we must find the functions that give the
distance traveled by Tika and by the duck in the first
t
seconds. These are just
antiderivatives of the velocity functions:
d (t) = 5t
,
T(t) = t
^{2}
. Since the duck gets a
100
foot head start, we should solve the equation
100 + 5t = t
^{2}
for
t
. The quadratic
formula gives
t = (5 + 5)/2
. Substituting into
T(t)
, we find that Tika must run
a total of about
164
feet.