Applications of Quadratic Functions

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 longdoes the apple take to reach your friend, and at what hei t above the ground does your friend catch it?

a. time to reach your friend ~ a.

b. height it was caught ~

name

-.58,,1It = Z-'--{ --=-l CD---:):--

t= {. 2

K?jF--- periOd_ .

b. h = -16( \.1 )2 + 38.4(\,,2.) + .96

h= 2..'1

2. Gretchen throws a ball straight up in the air. The quadratic function y = -16x2 +48x models the height in feet ofthe ball after x seconds. How long was the ball in the air? How long did it take the ball to reach the maximum heightbefore falling? Wbi;ltwas the maximum height? Ball Throw

Graph the information on the graplj,a. how long was ball in the air .25e(,.;b. time to reach maximum height II!; 5eGc. maximum height ~

3. Water is shot straight up out of a water soaker toy. The quadratic function y = -16x2 +32x models the height infeet of a water droplet after x seconds. How long is the water droplet in the air? What was the maximum height of thewater drop!et? Ho~ long did it take to reach this height? Water Soaker

~:~~~n~::~~~:~ droplet in air 25~ C _._._._...._.-;-- ---'---';--'-'-----'-':lszc

a. factor

-16x2 +48x = 0-1 ~)l(~ -3):. oto; () ex-:-37

b.x= -L{8zc.- (4t)x= \.5

b. time to reach its highest point

c. highest point J(0ft \i"-----------------.,---------------~----------~-------a. factor

-16x2 +32x = 0-( loi. (~ ..1.)-:,.D

i..";.0 [j.ill

b. c. substitute

x = ---=-3.~'l~2.L-1~)

x=--.:..l_

.J::g :-U;(I)t.f"32L,) f~ ., {(P

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

travels in feet can be modeled by the function d(t) = 5t2 +20t - 585 , where t is the time in seconds. To the nearestsecond, how long will it take her to reach the bottom? 95e c. 5 (-c2+1ft.-trr) -= 0 .

(t+I.?) - ~c-t:-I~ t::·

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 inthe air is given by the quadratic function h(t) = -16t2 +80t. x:. -:. -- eQ•.., f}Z; :1:5]

2.t-~ -

a. How high does the ball go? ~ what are you looking for? Vev=ie '/.b. for how many seconds is the ball in the air before it hits the ground? 5se.c what are finding? (a~is

- l Co t l t -5) = D. -/::.cri;--- '376. A record label uses the Slowing function to model the sales of a new release, y = -90t2 +81OOt,

whereas the number of albums sold is a function of, time, t, in days. On which day were the most albumssold? What is the maximum number of albums sold on that day?

a. which day were most albums sold 4. 5 b. maximum number of albums sold on that day )12.150what are youSmding? V e,y ~X- Lx-va, \It hat are you fmding? ~~

X-, .:~~<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 #qS.e~ what are you finding? ~ O() +SSHOW WORK!! NEATLY!! 'Z. (,0 rt» ill

-llo-t"2.+lo0::0 t:t t:~ -v f.'1It, t'Z.: -" 0 ,,-; ~ ;2 A./

- -lto ~t 1-:;48. 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 thetime in seconds. How many seconds will the fish fall before hitting the water?

Seconds to hit the water _{ (psee what are you looking for? J2.CD1-S-t"Z.+25~;:o .. /I (t+I~JSL""'()-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 javelinto hit the ground? t1 t:: '(]

Time to hit the ground l.:J 5ec. what are -you looking for? JL..00+$SHOW WORK NEATLY!! -. Oe-lZ.-\t 'i..LIe :.0 /l \{t 2; ,fJ"t;

~'L ': ~"i.&.fcS' -: t%t 1..5-.~ -.08

10. The barber's profitp each week depends on his charge c per haircut. It is modeled by the equationp = - 200c2 +2400c - 47 . t .ce should he charge for the largest profit? _e#~lo _

~, :2'"iOO_ '1'-.. {p2 (-2.6D)