Page 110 - Calculus Demystified

CHAPTER 3 Applications of the Derivative
EXAMPLE 3.16 97
A body is launched straight up from height 100 feet with some initial velocity.
It hitsthe ground after 10 seconds. What wasthat initial velocity?
SOLUTION
We are given that h 0 = 100. Thus
2
h(t) =−16t + v 0 t + 100.
Our job is to find v 0 . We also know that
2
0 = h(10) =−16 · 10 + v 0 · 10 + 100.
We solve this equation to find that v 0 = 150 ft /sec.

You Try It: On a certain planet, bodies fall with an acceleration due to gravity
2
of 10 ft/sec . A certain body is thrown down with an initial velocity of 5 feet per
second, and hits the surface 12 seconds later. From what height was it launched?

Exercises

2
1. Sketch the graph of f(x) = x/[x + 3], indicating all local maxima and
minima together with concavity properties.
2. What is the right circular cylinder of greatest volume that can be inscribed
upright in a right circular cone of radius 3 and height 6?
3. An air mattress (in the shape of a rectangular parallelepiped) is being inflated
in such a way that, at a given moment, its length is increasing by 1 inch
per minute, its width is decreasing by 0.5 inches per minute, and its height
is increasing by 0.3 inches per minute. At that moment its dimensions are

= 100 , w = 60 , and h = 15 . How is its volume changing at that time?

4. A certain body is thrown straight down at an initial velocity of 15 ft /sec. It
strikes the ground in 5 seconds. What is its initial height?
5. Because of viral infection, the shape of a certain cone-shaped cell is
changing. The height is increasing at the rate of 3 microns per minute.
For metabolic reasons, the volume remains constantly equal to 20 cubic
microns. At the moment that the radius is 5 microns, what is the rate of
change of the radius of the cell?
6. A silo is to hold 10,000 cubic feet of grain. The silo will be cylindrical
in shape and have a flat top. The floor of the silo will be the earth. What
dimensions of the silo will use the least material for construction?
7. Sketch the graph of the function g(x) = x·sin x. Show maxima and minima.