Greenwood press

90
QUADRATIC FUNCTIONS

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the ball to reach the ground. The equation then becomes 
0 = –0.5(9.8)(3.84)
2
+
v
o
(3.84) + 1.45, which has a solution of v
o
approximately equaling 18.4 meters
per second. Substituting this value into the general function will also provide
enough information to help you find the maximum height of your throw. 
The equation 
h = –0.5gt
2
+ v
o
t + h
o
can be simplified to 
h = –0.5gt
2
+ h
o
for objects in freefall because 
v
o
= 0 when an object is dropped. Therefore if you
plan to bungee-jump 200 meters off of a 250-meter-high bridge, then you should
expect to be dropping for about 6.4 seconds. This value comes from substituting
for the variables and solving the equation 
50 = –0.5(9.8)t
2
+ 250. (Note that the
ending position will be 50 meters above the ground, since the rope is only
extending 200 meters.) This general equation could also be used to estimate
heights and times for other objects that are released at high heights, such as the
steep drops on some amusement park rides.
Horizontal distance, such as the distance traveled after slamming on the brakes
in a car, can also be modeled with a quadratic function. In an effort to reconstruct
a traffic accident, a law office could use the function 
d = 0.02171v
2
+0.03576v
−0.24529 to determine how far a car could travel in feet, d, when breaking, or
how fast it was moving in feet per second, 
v, before it started braking. The law
office might also consider the average reaction time of 1.5 seconds upon seeing a
hazardous condition. So the total stopping distance, 
t, can be modeled with the
equation
t = 0.02171v
2
+ 0.03576v − 0.24529 + 1.5v, which simplifies to t =
0.02171v
2
+ 1.53576v − 0.24529.
Area applications can also be modeled by quadratic functions, because area
is represented in square units. For example, pizza prices depend on the amount
of pizza received, which is examining its area. However, on a pizza menu, the
sizes are revealed according to each pizza’s diameter. If a 12-inch pie costs $12,
a misconception would be to think that the 16-inch one should cost $16. A func-
tion to represent the price, 
p, of this type of pizza in terms of its diameter, d, is 
p = 0.106π(
d
2
)
2
, because it is a unit cost times the pizza’s area. The value 0.106 
is the price per square inch of pizza in dollars, assuming that the 12-inch pie for
$12 will have the same unit-cost value as any other size pizza. Therefore a 16- 
inch pizza should cost 
p = 0.106π(
16
2
)
2
≈ $21.31. The restaurant, however, may 
decide to give a financial incentive for customers to purchase larger pies and
reduce this price to somewhere near $20.
Devising and purchasing tin cans for food are applications of surface area
that can be represented by a quadratic function. Since most tin cans are cylindri-
cal, the surface area can be determined by finding the area of the rectangular lat-
eral area and the sum of the two bases, as shown in the following figure. If the
manufacturer determines the height of its cans to be 4 inches tall with a variable
radius, then the amount of sheet metal in square inches, 
a, needed for each can
would be 
a = 8πr + 2πr
2
, where 
r is the radius of the can in inches.
If tin costs the manufacturer $0.003 per square inch, then the materials cost,
c, to produce each case of twenty-four cans can be represented by the function
c = (24)(0.003)a = (24)(0.003)(8πr + 2πr
2
), which simplifies to c ≈ 1.81r+