Calculus 220, section 2.5 Optimization Problems (Applications)
notes by Tim Pilachowski
Oh boy—now we get to start on word problems! First, a definition: optimize – (verb) to make as perfect or
effective as possible. In calculus terms, for anything optimal, we will be searching for some sort of maximum or
minimum.
Example A: The function calculates the height of an object, s, after time, t, thrown
upward at 10 feet per second from a bridge which is 240 feet above the river below. In section 1.8 lecture notes
we asked:
()
2401016 2++−= ttts
a) What is the height of the rock after 2 seconds? answer: Find
(
)
1962
=
s feet.
b) How much height did the rock gain after 2 seconds? answer: Calculate
(
)
(
)
=
−
02 ss loss of 44 feet of height.
c) What is the velocity of the rock after 2 seconds? answer: Find
(
)
=
′
2sthe rock is falling at 54 feet per second.
d) What is the average velocity during the first 2 seconds? answer Find
(
)
(
)
=
−
−
02
02 ss on average the rock has
fallen at 22 feet per second.
New questions:
e) How long does it take for the rock to reach its maximum height? Answer: 16
5 sec.
f) What is the maximum height above the water reached by the rock? Answer: 16
5
241 feet
g) What is the maximum height above the bridge reached by the rock? Answer: 16
5
1 feet
