Applications of Quadratic Functions
name
K?jF---
periOd_ .
1. Suppose you are tossing an apple up to a friend on a third-story balcony. After t seconds, the height ofthe apple in feet
is given by
h(t)
= -16t2
+ 38.4t + 0.96 . Your
friend catches the apple just as it reaches its highest point. How long
does the apple take to reach your friend, and at what hei
t above the ground does your friend catch it?
-.58,,1I
a. time to reach your friend ~
a.
b. height it was caught ~
t
= Z-'--{ --=-l CD---:):--
t=
{.
h = -16(
b.
2
2. Gretchen throws a ball straight up in the air. The quadratic function
h=
y
\.1 )2 + 38.4(\,,2.) + .96
2..'1
= -16x2 + 48x
models the height in feet of
the ball after x seconds. How long was the ball in the air? How long did it take the ball to reach the maximum height
before falling? Wbi;ltwas the maximum height?
Ball Throw
Graph the information on the graplj,
a. how long was ball in the air
.25e(,.;
II!; 5eG
b. time to reach maximum height
c. maximum height ~
a. factor
2
-16x
b.
+ 48x
-1 ~)l(~
x= -L{8
=0
zc.- (4t)
x= \.5
-3):. o
to;() ex-:- 37
3. Water is shot straight up out of a water soaker toy. The quadratic function
y = -16x2 + 32x
models the height in
feet of a water droplet after x seconds. How long is the water droplet in the air? What was the maximum height of the
water drop!et? Ho~ long did it take to reach this height?
Water Soaker
~:~~~n~::~~~:~
droplet in air
b. time to reach its highest point
c. highest point
25~ C
lszc
_._._._
....
_.-;-- ---'---';--''-----'-':
J(0 ft
-----------------.,---------------~----------~------a. factor
2
-16x
b.
+ 32x
=0
-( loi. (~ ..1.)-:,. D
i..";.0 [j.ill
c. substitute
x
=
\i"
.J::
---=-3.~'l~ g :-U;(I)t.f"32L,) f
2.L-1 ~)
~ ., {(P
x=--.:..l_
Time (Seconds)
4. A bicyclist is riding at a speed of20 milh when she starts down a long hill which is 585 ft long. The distance d she
d(t)
travels in feet can be modeled by the function
= 5t2 + 20t -
9 5e c.
second, how long will it take her to reach the bottom?
585 , where
t is the time in seconds. To the nearest
2 1ft.
0
.
5 (-c +
-trr) -=
(t+I.?)
- ~c
t::·
-t:-I~
5. Suppose you throw a ball straight up from the ground with a velocity of 80 ftls. s the ball moves upward,
gravity slows it. Eventually the ball begins to fall back to the ground. The height h of the ball after t seconds in
2
+ 80t.
the air is given by the quadratic function h(t) = -16t
a. How high does the ball go? ~
f}Z; :1:5]
x:. -:.--2.t-~
eQ•.., -
V ev=ie '/.
what are you looking for?
b. for how many seconds is the ball in the air before it hits the ground?
- l Co t l t -5) = D
5se.c
what are finding?
. -/::.cr i;--- '37
2
+ 81OOt,
function to model the sales of a new release, y = -90t
6. A record label uses the Slowing
(a~is
whereas the number of albums sold is a function of, time, t, in days. On which day were the most albums
sold? What is the maximum number of albums sold on that day?
4. 5
a. which day were most albums sold
what are you Smding?
X-,
b. maximum number of albums sold on that day
V e,y ~X- Lx-va, \It
)12.150
hat are you fmding? ~~
.:~~<fo)
Q?:~LfS)
~: -qO(L\S}1. +~IM(45)
7. If a tightrope walker falls, he will land on a safety net. His height h in feet after a fall can be modeled by
h(t) = 60 -16t2
,
where t is the time in seconds. How many seconds will the tightrope walker fall before
landing on the safety net?
l # q S.e~ what are you finding?
SHOW WORK!! NEATLY!!
'Z.
t:t
-llo-t"2.+lo0::0
It, t'Z.: -" 0
-lto
,,-;
~t
~ O() +S
t:~rt»
(,0
;2
~
1-:;4
-v
ill
f.'1
A./
8. A bald eagle snatc es a fish from a lake and flies to ~ altitude of256 ft. The fish manages to squirm free
and falls back down into the lake. Its height h in feet can be modeled by h(t) = 256 - t2 ,where t is the
time in seconds. How many seconds will the fish fall before hitting the water?
(p see
what are you looking for? J2. CD 1-S
-t"Z.+25~;:o .. /I (t+I~JSL""'()
Seconds to hit the water _{
-1 Ct 2.-~5(0) ~O
<
to;. :-1 f.t,
~
~ (.
2
.
9. The quadratic function that approximates the height 0
ill
ow IS h(t) = -0.08t +4.48 where tIS
the time in seconds after it is thrown and h is the javelin's height I fee\. How long will it take for the javelin
to hit the ground?
t1 t::
'(]
Time to hit the ground
l.:J
what are -you looking for? JL.. 00+$
5ec.
Oe-lZ.-\t 'i..
LIe :. 0
SHOW WORK NEATLY!! -.
~'L
':
-.~
~"i.&.fcS' -:
/l
\{t 2; ,fJ"t;
t%t 1..5
-.08
10. The barber's profitp each week depends on his charge c per haircut. It is modeled by the equation
p
= - 200c2 + 2400c -
~,
:2'"iOO_
2 (-2.6D)
47
.
t
'1'-.. {p
.ce should he charge for the largest profit?
_e#~lo _