The flight path of a rocket, launched from the ground, is modeled by y = 0.25(40x-x²), where y is the height of the rocket in feet from the ground and x is the time in seconds after it was launched. Find the vertex of the parabola and explain what it represents in terms of the context. O (40, 0); rocket lands at 0 feet after 40 seconds O (20, 100); rocket reaches maximum height of 20 feet after 100 seconds O (40, 0); rocket reaches 40 feet after 0 seconds O (20, 100); rocket reaches maximum height 100 feet after 20 seconds
The flight path of a rocket, launched from the ground, is modeled by y = 0.25(40x-x²), where y is the height of the rocket in feet from the ground and x is the time in seconds after it was launched. Find the vertex of the parabola and explain what it represents in terms of the context. O (40, 0); rocket lands at 0 feet after 40 seconds O (20, 100); rocket reaches maximum height of 20 feet after 100 seconds O (40, 0); rocket reaches 40 feet after 0 seconds O (20, 100); rocket reaches maximum height 100 feet after 20 seconds
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
Related questions
Question
![**Understanding the Rocket's Flight Path**
The flight path of a rocket, launched from the ground, is modeled by the equation:
\[ y = 0.25(40x - x^2) \]
In this equation:
- \( y \) represents the height of the rocket (in feet) from the ground.
- \( x \) represents the time (in seconds) after the rocket has been launched.
**Question:**
Find the vertex of the parabola represented by the equation and explain what it represents in terms of the rocket's flight.
**Answer Choices:**
1. **(40, 0); rocket lands at 0 feet after 40 seconds**
2. **(20, 100); rocket reaches maximum height of 20 feet after 100 seconds**
3. **(40, 0); rocket reaches 40 feet after 0 seconds**
4. **(20, 100); rocket reaches maximum height of 100 feet after 20 seconds**
**Explanation:**
- To find the vertex of the parabola, consider the given quadratic equation \( y = 0.25 (40x - x^2) \).
- The standard form for a quadratic equation is \( y = ax^2 + bx + c \). Here, it's helpful to rewrite the given equation into that standard form:
\[ y = 0.25(-x^2 + 40x) \]
- The vertex form of a parabola is given by \( x = -\frac{b}{2a} \), where \( a \) and \( b \) are coefficients from the standard form equation \( ax^2 + bx + c \).
- In this case, \( a = -0.25 \) and \( b = 10 \).
- Calculate the vertex:
\[ x = -\frac{b}{2a} = -\frac{40}{2(-1)} = 20 \]
- Substituting \( x = 20 \) back into the equation to find the maximum height \( y \):
\[ y = 0.25 (40 \times 20 - 20^2) \]
\[ y = 0.25 (800 - 400) \]
\[ y = 0.25 \times 400 \]
\[ y = 100 \]
Therefore, the vertex of the parabola is at \( (20, 100) \), meaning](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3e08c0ed-2478-4250-8ecc-80933702ca06%2Ffccb1145-9697-4fd0-ad5a-95cc5052d512%2Fc2d5lmc_processed.png&w=3840&q=75)
Transcribed Image Text:**Understanding the Rocket's Flight Path**
The flight path of a rocket, launched from the ground, is modeled by the equation:
\[ y = 0.25(40x - x^2) \]
In this equation:
- \( y \) represents the height of the rocket (in feet) from the ground.
- \( x \) represents the time (in seconds) after the rocket has been launched.
**Question:**
Find the vertex of the parabola represented by the equation and explain what it represents in terms of the rocket's flight.
**Answer Choices:**
1. **(40, 0); rocket lands at 0 feet after 40 seconds**
2. **(20, 100); rocket reaches maximum height of 20 feet after 100 seconds**
3. **(40, 0); rocket reaches 40 feet after 0 seconds**
4. **(20, 100); rocket reaches maximum height of 100 feet after 20 seconds**
**Explanation:**
- To find the vertex of the parabola, consider the given quadratic equation \( y = 0.25 (40x - x^2) \).
- The standard form for a quadratic equation is \( y = ax^2 + bx + c \). Here, it's helpful to rewrite the given equation into that standard form:
\[ y = 0.25(-x^2 + 40x) \]
- The vertex form of a parabola is given by \( x = -\frac{b}{2a} \), where \( a \) and \( b \) are coefficients from the standard form equation \( ax^2 + bx + c \).
- In this case, \( a = -0.25 \) and \( b = 10 \).
- Calculate the vertex:
\[ x = -\frac{b}{2a} = -\frac{40}{2(-1)} = 20 \]
- Substituting \( x = 20 \) back into the equation to find the maximum height \( y \):
\[ y = 0.25 (40 \times 20 - 20^2) \]
\[ y = 0.25 (800 - 400) \]
\[ y = 0.25 \times 400 \]
\[ y = 100 \]
Therefore, the vertex of the parabola is at \( (20, 100) \), meaning
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 1 images
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning