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.