I need help solving all parts of question 65 parts a-f included in the two attachments.

Please note that this is not graded work. I obtained this question from an old text book to help me practice problem sets. Do let me know if you have additional questions.
Transcribed Image Text:186 CHAPTER 3 Polynomial Functions Curve Fitting Find the equation of the quadratic function satisfying the given conditions. (Hint: Determine values of a, h, and k that satisfy P(x) = a(x - h)2 + k.) Express your answer in the form P(x) = ax² + bx + c. Use your calculator to support your results. 55. Vertex (-1,-4); through (5, 104) 56. Vertex (-2, -3); through (0, -19) 57. Vertex (8, 3); through (10, 5) 58. Vertex (-6, -12); through (6, 24) 59. Vertex (-4,-2); through (2, -26) 60. Vertex (5, 6); through (1,-6) (Modeling) Solve each problem. 61. Heart Rate An athlete's heart rate R in beats per minute after x minutes is given by Arfon R(x) = 2(x-4)² + 90, where 0 ≤ x ≤ 8. (a) Describe the heart rate during this period of time. (b) Determine the minimum heart rate during this 8-minute period. 62. Social Security Assets The graph shows how Social Security assets are expected to change as the number of retirees receiving benefits increases. Social Security Assets* Billions of Dollars 4000 3000 2000 1000 0 ***** 2010 1 2015 UND AV 2020 Year *Projected Source: Social Security Administration 2025 2030 2035 The graph suggests that a quadratic function would be a good fit to the data, which are approximated by f(x) = -10.36x² + 431.8x - 650. In the model, x = 10 represents 2010, x = 15 represents 2015, and so on, and f(x) is in billions of dollars. (a) Explain why the coefficient of x² in the model is negative, based on the graph. o this (b) Analytically determine the vertex of the graph. (c) Interpret the answer to part (b) as it relates t application. 737 weight 63. Heart Rate The heart rate of an athlete while training is recorded for 4 minutes. The table lists the hear rate after x minutes. Time (min) Heart rate (bpm) Sitzworz 1 0 84 111 2 20 ft 300 ft 3 120 110 buloa last vitam woh (a) Explain why the data are not linear. (b) Find a quadratic function f that models the data. (c) What is the domain of your function? 64. Suspension Bridge The cables that support a suspension bridge, such as the Golden Gate Bridge, can be modeled by parabolas. Suppose that a 300-foot-long suspension bridge has at its ends towers that are 120 feet tall, as shown in the figure. If the cable comes within 20 feet of the road at the center of the bridge, find a function that models the height of the cable above the road a distance of x feet from the center of the bridge. $215/70 4 85 (Modeling) The formula for the height of a projectile is s(t) = -16t² + vot + So, d s(t) is is in feet. 120 ft where t is time in seconds, so is the initial height in feet, vo is the initial velocity in feet per second, and this formula to solve Exercises 65-68. Use 65. Height of a Projected Rock A rock is launched upward from ground level with an initial velocity of 90 feet per second. Let t represent the amount of time elapsed after it is launched. (a) Explain why t cannot be a negative number in this situ ation. (b) Explain why so = 0 in this problem. (c) Give the functions that describes the height of the rock as a function of t. (d) How high will the rock b launched? (e) What is the maximum heig After how many seconds mine the answer analyticall (f) After how many seconds ground? Determine the ansv 66. Height of a Toy Rocket A toy the top of a building 50 feet tall 200 feet per second. Let t repre elapsed after the launch. (a) Express the height s as a fun (b) Determine both analytically at which the rocket reaches high will it be at that time? (c) For what time interval will t 300 feet above ground level" graphically, and give times t second. (d) After how many seconds will th Determine the answer graphic: 67. Height of a Projected Ball A b from ground level with an initial second. moubo est (a) Determine graphically whethe height of 355 feet. If it will, when this happens. If it will m a graphical interpretation. on hi nerode Si vorb.0 = (12) v 3.3 Quadratic Ec Zero-Product Property Square Root Prop Quadratic Equations Solving Quadratic 0 A sns pl FUC
Transcribed Image Text:the graph. as it relates to this athlete while weight ne table lists the heart 3 110 4 85 r. odels the data. on? upport a suspension can be modeled by g suspension bridge et tall, as shown in O feet of the road at on that models the cance of x feet from 120 ft rojectile is X ght in feet, vo is is in feet. Use T aunched upward y of 90 feet per e elapsed after it mber in this situ- eight of the rock (d) How high will the rock be 1.5 seconds after it is launched? (e) What is the maximum height attained by the rock? After how many seconds will this happen? Deter- mine the answer analytically and graphically. (f) After how many seconds will the rock hit the ground? Determine the answer graphically. ps 66. Height of a Toy Rocket A toy rocket is launched from the top of a building 50 feet tall at an initial velocity of 200 feet per second. Let t represent the amount of time elapsed after the launch. (a) Determine graphically whether the ball will reach a height of 355 feet. If it will, determine the time(s) when this happens. If it will not, explain why, using a graphical interpretation. SUOR noituto2 solisluplc ● egal (a) Express the height s as a function of the time t. (b) Determine both analytically and graphically the time at which the rocket reaches its highest point. How high will it be at that time? (c) For what time interval will the rocket be more than 300 feet above ground level? Determine the answer graphically, and give times to the nearest tenth of a second. (d) After how many seconds will the rocket hit the ground? Determine the answer graphically. 10001 ubong-018 Concept Check Sketch a graph of a quadratic function that 67. Height of a Projected Ball A ball is launched upward fon po satisfies each set of given conditions. Use symmetry to label from ground level with an initial velocity of 150 feet per ner another point on your graph. second. 69. Vertex (-2,-3); through (1,4) 70. Vertex (5, 6); through (1, -6) 71. Maximum value of 1 at x = 3; y-intercept is (0, -4) 72. Minimum value of -4 at x = -3; y-intercept is (0, 3) ES ● 3.3 Quadratic Equations and Inequalities (b) Repeat part (a) for a ball launched from a height of 30 feet with an initial velocity of 250 feet per second. 68. Height of a Projected Ball on the Moon An astronaut on the moon throws a baseball upward. The astronaut is 6 feet, 6 inches tall and the initial velocity of the ball is 30 feet per second. The height of the ball is approximated by the function 0 s(t) = -2.7t² + 30t + 6.5, hamma.o where t is the number of seconds after the ball was thrown. (a) After how many seconds is the ball 12 feet above the moon's surface? (b) How many seconds after it is thrown will the ball Top return to the surface? (c) The ball will never reach a height of 100 feet. How can this be determined analytically? 3.3 Quadratic Equations and Inequalities gloz Zero-Product Property Square Root Property and Completing the Square Quadratic Formula and the Discriminant Solving Quadratic Equations Solving Quadratic Inequalities Formulas Involving Quadratics 187 Joc A quadratic equation is defined as follows. TE noitulo2 piylená to dqszy od to ● Quadratic Equation in One Variable An equation that can be written in the form ax² + bx + c = 0, where a, b, and c are real numbers, with a # 0, is a quadratic equation in standard form.

Question

I need help solving all parts of question 65 parts a-f included in the two attachments.

Please note that this is not graded work. I obtained this question from an old text book to help me practice problem sets. Do let me know if you have additional questions.

186 CHAPTER 3 Polynomial Functions
Curve Fitting Find the equation of the quadratic function
satisfying the given conditions. (Hint: Determine values of a,
h, and k that satisfy P(x) = a(x - h)2 + k.) Express your
answer in the form P(x) = ax² + bx + c. Use your
calculator to support your results.
55. Vertex (-1,-4); through (5, 104)
56. Vertex (-2, -3); through (0, -19)
57. Vertex (8, 3); through (10, 5)
58. Vertex (-6, -12); through (6, 24)
59. Vertex (-4,-2); through (2, -26)
60. Vertex (5, 6); through (1,-6)
(Modeling) Solve each problem.
61. Heart Rate An athlete's heart rate R in beats per minute
after x minutes is given by
Arfon
R(x) = 2(x-4)² + 90,
where 0 ≤ x ≤ 8.
(a) Describe the heart rate during this period of time.
(b) Determine the minimum heart rate during this 8-minute
period.
62. Social Security Assets The graph shows how Social
Security assets are expected to change as the number of
retirees receiving benefits increases.
Social Security Assets*
Billions of Dollars
4000
3000
2000
1000
0
*****
2010
1
2015
UND
AV
2020
Year
*Projected
Source: Social Security Administration
2025
2030
2035
The graph suggests that a quadratic function would be a good
fit to the data, which are approximated by
f(x) = -10.36x² + 431.8x - 650.
In the model, x = 10 represents 2010, x = 15 represents 2015,
and so on, and f(x) is in billions of dollars.
(a) Explain why the coefficient of x² in the model is negative,
based on the graph.
o this
(b) Analytically determine the vertex of the graph.
(c) Interpret the answer to part (b) as it relates t
application.
737
weight
63. Heart Rate The heart rate of an athlete while
training is recorded for 4 minutes. The table lists the hear
rate after x minutes.
Time (min)
Heart rate
(bpm)
Sitzworz
1
0
84 111
2
20 ft
300 ft
3
120 110
buloa last vitam woh
(a) Explain why the data are not linear.
(b) Find a quadratic function f that models the data.
(c) What is the domain of your function?
64. Suspension Bridge The cables that support a suspension
bridge, such as the Golden Gate Bridge, can be modeled by
parabolas. Suppose that a 300-foot-long suspension bridge
has at its ends towers that are 120 feet tall, as shown in
the figure. If the cable comes within 20 feet of the road at
the center of the bridge, find a function that models the
height of the cable above the road a distance of x feet from
the center of the bridge.
$215/70
4
85
(Modeling) The formula for the height of a projectile is
s(t) = -16t² + vot + So,
d s(t) is
is in feet.
120 ft
where t is time in seconds, so is the initial height in feet, vo is
the initial velocity in feet per second, and
this formula to solve Exercises 65-68.
Use
65. Height of a Projected Rock A rock is launched upward
from ground level with an initial velocity of 90 feet per
second. Let t represent the amount of time elapsed after it
is launched.
(a) Explain why t cannot be a negative number in this situ
ation.
(b) Explain why so = 0 in this problem.
(c) Give the functions that describes the height of the rock
as a function of t.
(d) How high will the rock b
launched?
(e) What is the maximum heig
After how many seconds
mine the answer analyticall
(f) After how many seconds
ground? Determine the ansv
66. Height of a Toy Rocket A toy
the top of a building 50 feet tall
200 feet per second. Let t repre
elapsed after the launch.
(a) Express the height s as a fun
(b) Determine both analytically
at which the rocket reaches
high will it be at that time?
(c) For what time interval will t
300 feet above ground level"
graphically, and give times t
second.
(d) After how many seconds will th
Determine the answer graphic:
67. Height of a Projected Ball A b
from ground level with an initial
second.
moubo
est
(a) Determine graphically whethe
height of 355 feet. If it will,
when this happens. If it will m
a graphical interpretation.
on hi nerode
Si vorb.0 = (12) v
3.3 Quadratic Ec
Zero-Product Property Square Root Prop
Quadratic Equations Solving Quadratic
0
A
sns pl
FUC

the graph.
as it relates to this
athlete while weight
ne table lists the heart
3
110
4
85
r.
odels the data.
on?
upport a suspension
can be modeled by
g suspension bridge
et tall, as shown in
O feet of the road at
on that models the
cance of x feet from
120 ft
rojectile is
X
ght in feet, vo is
is in feet. Use
T
aunched upward
y of 90 feet per
e elapsed after it
mber in this situ-
eight of the rock
(d) How high will the rock be 1.5 seconds after it is
launched?
(e) What is the maximum height attained by the rock?
After how many seconds will this happen? Deter-
mine the answer analytically and graphically.
(f) After how many seconds will the rock hit the
ground? Determine the answer graphically. ps
66. Height of a Toy Rocket A toy rocket is launched from
the top of a building 50 feet tall at an initial velocity of
200 feet per second. Let t represent the amount of time
elapsed after the launch.
(a) Determine graphically whether the ball will reach a
height of 355 feet. If it will, determine the time(s)
when this happens. If it will not, explain why, using
a graphical interpretation.
SUOR
noituto2 solisluplc
●
egal
(a) Express the height s as a function of the time t.
(b) Determine both analytically and graphically the time
at which the rocket reaches its highest point. How
high will it be at that time?
(c) For what time interval will the rocket be more than
300 feet above ground level? Determine the answer
graphically, and give times to the nearest tenth of a
second.
(d) After how many seconds will the rocket hit the ground?
Determine the answer graphically.
10001 ubong-018
Concept Check Sketch a graph of a quadratic function that
67. Height of a Projected Ball A ball is launched upward fon po satisfies each set of given conditions. Use symmetry to label
from ground level with an initial velocity of 150 feet per
ner another point on your graph.
second.
69. Vertex (-2,-3); through (1,4)
70. Vertex (5, 6); through (1, -6)
71. Maximum value of 1 at x = 3; y-intercept is (0, -4)
72. Minimum value of -4 at x = -3; y-intercept is (0, 3)
ES
●
3.3 Quadratic Equations and Inequalities
(b) Repeat part (a) for a ball launched from a height
of 30 feet with an initial velocity of 250 feet per
second.
68. Height of a Projected Ball on the Moon An astronaut
on the moon throws a baseball upward. The astronaut is
6 feet, 6 inches tall and the initial velocity of the ball is
30 feet per second. The height of the ball is approximated
by the function
0
s(t) = -2.7t² + 30t + 6.5,
hamma.o
where t is the number of seconds after the ball was
thrown.
(a) After how many seconds is the ball 12 feet above the
moon's surface?
(b) How many seconds after it is thrown will the ball
Top return to the surface?
(c) The ball will never reach a height of 100 feet. How
can this be determined analytically?
3.3 Quadratic Equations and Inequalities
gloz
Zero-Product Property Square Root Property and Completing the Square Quadratic Formula and the Discriminant Solving
Quadratic Equations Solving Quadratic Inequalities Formulas Involving Quadratics
187
Joc
A quadratic equation is defined as follows.
TE
noitulo2 piylená
to dqszy od to
●
Quadratic Equation in One Variable
An equation that can be written in the form
ax² + bx + c = 0,
where a, b, and c are real numbers, with a # 0, is a quadratic equation in
standard form.

Accepted Answer

The rock is launched upward.
Initial velocity of rock = u = 90feet/s
t = time after rock was…