Height of a Projected Rock A rock is launched upwa from ground level with an initial velocity of 90 feet p second. Let t represent the amount of time elapsed after s launched. a) Explain why I cannot be a negative number in this si ation. b) Explain why so = 0 in this problem. =) Give the functions that describes the height of the rod as a function of 1.
Height of a Projected Rock A rock is launched upwa from ground level with an initial velocity of 90 feet p second. Let t represent the amount of time elapsed after s launched. a) Explain why I cannot be a negative number in this si ation. b) Explain why so = 0 in this problem. =) Give the functions that describes the height of the rod as a function of 1.
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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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.
![### Polynomial Functions
#### Curve Fitting
- **Objective:** Find the equation of the quadratic function that meets the given conditions.
- **Method:** Determine values of \( a \), \( b \), and \( k \) such that \( f(x) = a(x - h)^2 + k \). Express the function as \( f(x) = ax^2 + bx + c \). Use a calculator to support your results.
**Problems:**
- 55. Vertex (-1, -4); through (3, 104)
- 56. Vertex (-2, 3); through (0, -19)
- 57. Vertex (3, 8); 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 - Problem Solving
61. **Heart Rate**
- Function: The heart rate \( R \) in beats per minute after \( x \) minutes is given by:
\[
R(x) = 20x^2 - 49x + 90
\]
- Domain: \( 0 \leq x \leq 8 \)
- Tasks:
- (a) Describe the heart rate during this period.
- (b) Determine the minimum heart rate during this 8-minute period.
62. **Social Security**
- Description: The graph displays the projected change in Social Security assets due to changes in retiree numbers.
- Graph: **Social Security Assets*** in billions of dollars over years (2010, 2015, 2020, 2025, 2030, 2035).
- Equation: \( f(x) = -10.362x^2 + 431.8x - 650 \)
- Tasks:
- (a) In this model, \( x = 10 \) for 2010; \( x = 15 \) for 2015.
- (b) Explain why the coefficient of \( x^2 \) is negative.
63. **Heart Rate Table**
- The table records the heart rate of an athlete post-training over 4 minutes:
\[
\begin{array}{|](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F233dd85c-372f-4e28-8989-94d95de9925b%2F7e5c89cf-f116-4126-8f46-72653ec6f6b0%2Fggzgovm_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Polynomial Functions
#### Curve Fitting
- **Objective:** Find the equation of the quadratic function that meets the given conditions.
- **Method:** Determine values of \( a \), \( b \), and \( k \) such that \( f(x) = a(x - h)^2 + k \). Express the function as \( f(x) = ax^2 + bx + c \). Use a calculator to support your results.
**Problems:**
- 55. Vertex (-1, -4); through (3, 104)
- 56. Vertex (-2, 3); through (0, -19)
- 57. Vertex (3, 8); 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 - Problem Solving
61. **Heart Rate**
- Function: The heart rate \( R \) in beats per minute after \( x \) minutes is given by:
\[
R(x) = 20x^2 - 49x + 90
\]
- Domain: \( 0 \leq x \leq 8 \)
- Tasks:
- (a) Describe the heart rate during this period.
- (b) Determine the minimum heart rate during this 8-minute period.
62. **Social Security**
- Description: The graph displays the projected change in Social Security assets due to changes in retiree numbers.
- Graph: **Social Security Assets*** in billions of dollars over years (2010, 2015, 2020, 2025, 2030, 2035).
- Equation: \( f(x) = -10.362x^2 + 431.8x - 650 \)
- Tasks:
- (a) In this model, \( x = 10 \) for 2010; \( x = 15 \) for 2015.
- (b) Explain why the coefficient of \( x^2 \) is negative.
63. **Heart Rate Table**
- The table records the heart rate of an athlete post-training over 4 minutes:
\[
\begin{array}{|
![**3.3 Quadratic Equations and Inequalities**
### Problems and Exercises
**64.** A number of questions are provided around the theme of projectile motion, involving scenarios with rockets and balls being launched from varying heights and speeds. These questions require both analytical and graphical methods to solve for aspects such as height at different times, maximum height, and time until hitting the ground.
**66. Height of a Toy Rocket**
A toy rocket is launched from a height of 50 feet with an initial velocity of 200 feet per second. Let \( t \) represent the time elapsed after the launch.
- **(a)** Express the height as a function of time \( t \).
- **(b)** Determine both analytically and graphically the time at which the rocket reaches its highest point. Calculate and state the height at this time.
- **(c)** Find the time when the rocket will reach a height above 3600 feet. Determine the answer graphically to the nearest tenth of a second.
- **(d)** Calculate how many seconds until the rocket hits the ground. Represent the answer graphically.
**67. Height of a Projected Ball**
A ball is launched with an initial velocity of 150 feet per second from ground level.
- **(a)** Determine if the ball will reach a height of 355 feet. Analyze graphically and find the time when this occurs, or explain using graphical interpretation if it doesn't happen.
**68. Height of a Projected Ball on the Moon**
An astronaut on the moon throws a baseball upward with an initial height of 6 feet and an initial velocity of 30 feet per second. The height \( s(t) \) of the ball as a function of time is:
\[ s(t) = -2.7t^2 + 30t + 6.5 \]
- Where \( t \) is the time in seconds after the ball was thrown.
Graphically explore:
- **(b)** The time in seconds when the ball is 12 feet above the moon's surface.
- **(c)** How many seconds after launching the ball will it return to the surface, supported by analytical interpretation.
- **(e)** Whether the ball reaches a height of 100 feet, and the reasoning behind it analytically.
### Concept Check
Instructions to sketch the graph of a quadratic function that meets specified conditions, utilizing symmetry to label an additional point on the graph.
### Mathematics and](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F233dd85c-372f-4e28-8989-94d95de9925b%2F7e5c89cf-f116-4126-8f46-72653ec6f6b0%2Fzkn1a6_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**3.3 Quadratic Equations and Inequalities**
### Problems and Exercises
**64.** A number of questions are provided around the theme of projectile motion, involving scenarios with rockets and balls being launched from varying heights and speeds. These questions require both analytical and graphical methods to solve for aspects such as height at different times, maximum height, and time until hitting the ground.
**66. Height of a Toy Rocket**
A toy rocket is launched from a height of 50 feet with an initial velocity of 200 feet per second. Let \( t \) represent the time elapsed after the launch.
- **(a)** Express the height as a function of time \( t \).
- **(b)** Determine both analytically and graphically the time at which the rocket reaches its highest point. Calculate and state the height at this time.
- **(c)** Find the time when the rocket will reach a height above 3600 feet. Determine the answer graphically to the nearest tenth of a second.
- **(d)** Calculate how many seconds until the rocket hits the ground. Represent the answer graphically.
**67. Height of a Projected Ball**
A ball is launched with an initial velocity of 150 feet per second from ground level.
- **(a)** Determine if the ball will reach a height of 355 feet. Analyze graphically and find the time when this occurs, or explain using graphical interpretation if it doesn't happen.
**68. Height of a Projected Ball on the Moon**
An astronaut on the moon throws a baseball upward with an initial height of 6 feet and an initial velocity of 30 feet per second. The height \( s(t) \) of the ball as a function of time is:
\[ s(t) = -2.7t^2 + 30t + 6.5 \]
- Where \( t \) is the time in seconds after the ball was thrown.
Graphically explore:
- **(b)** The time in seconds when the ball is 12 feet above the moon's surface.
- **(c)** How many seconds after launching the ball will it return to the surface, supported by analytical interpretation.
- **(e)** Whether the ball reaches a height of 100 feet, and the reasoning behind it analytically.
### Concept Check
Instructions to sketch the graph of a quadratic function that meets specified conditions, utilizing symmetry to label an additional point on the graph.
### Mathematics and
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