College Pranks. A student uses rubber tubing to launch a water balloon from the roof of his dormitory. The height h (in feet) of the balloon, t seconds after being launched, is given by the formula h = -16t 2 + 48t + 64. After how many seconds will the balloon hit the ground? When the water balloon hits the ground, its height will be 0 feet.To find the time that it takes for the balloon to hit the ground, we set h equal to 0, and solve the quadratic equation for t. h = -16t 2 + 48t + 64 0 = -16t 2 + 48t + 64 Substitute 0 for the height, h. This is a quadratic equation. 0 = -16t 2 + 48t + 64 0 = -16(t2 - 3t - 0 = -16(t + Factor out the opposite of the GCF, -16. )(t – 4) Factor the trinomial. t -[ | = 0 Since -16 cannot equal 0, discard that possibility. Set each factor that contains a variable equal to 0. t + = 0 or 1 = -1 t = 4 Solve each equation. The equation has two solutions, -1 and 4. Since t represents time, and, in this case, time cannot be negative, we discard -1. The second solution, 4, indicates that the balloon hits the ground 4 second after being launched. Check this result by substituting 4 for t in h = -16t 2 + 48t + 64. You should get h =
College Pranks. A student uses rubber tubing to launch a water balloon from the roof of his dormitory. The height h (in feet) of the balloon, t seconds after being launched, is given by the formula h = -16t 2 + 48t + 64. After how many seconds will the balloon hit the ground? When the water balloon hits the ground, its height will be 0 feet.To find the time that it takes for the balloon to hit the ground, we set h equal to 0, and solve the quadratic equation for t. h = -16t 2 + 48t + 64 0 = -16t 2 + 48t + 64 Substitute 0 for the height, h. This is a quadratic equation. 0 = -16t 2 + 48t + 64 0 = -16(t2 - 3t - 0 = -16(t + Factor out the opposite of the GCF, -16. )(t – 4) Factor the trinomial. t -[ | = 0 Since -16 cannot equal 0, discard that possibility. Set each factor that contains a variable equal to 0. t + = 0 or 1 = -1 t = 4 Solve each equation. The equation has two solutions, -1 and 4. Since t represents time, and, in this case, time cannot be negative, we discard -1. The second solution, 4, indicates that the balloon hits the ground 4 second after being launched. Check this result by substituting 4 for t in h = -16t 2 + 48t + 64. You should get h =
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,...
Related questions
Question
![### College Pranks: Calculating the Time for a Water Balloon to Hit the Ground
A student uses rubber tubing to launch a water balloon from the roof of his dormitory. The height \( h \) (in feet) of the balloon, \( t \) seconds after being launched, is given by the formula
\[ h = -16t^2 + 48t + 64. \]
After how many seconds will the balloon hit the ground?
To find out when the water balloon hits the ground, we need to determine when the height \( h \) will be 0 feet. So, we set \( h \) equal to 0 and solve the quadratic equation for \( t \).
#### Step-by-Step Solution
1. **Set the height to 0** (because the balloon hits the ground when its height is 0):
\[ h = -16t^2 + 48t + 64 \]
\[ 0 = -16t^2 + 48t + 64 \]
2. **Factor the quadratic equation**:
\[ 0 = -16(t^2 - 3t - 4) \]
\[ 0 = -16(t + 1)(t - 4) \]
- Factor out the opposite of the Greatest Common Factor (GCF), which is -16.
- Factor the trinomial \( t^2 - 3t - 4 \).
3. **Solve for \( t \)** by setting each factor that contains a variable to 0:
\[ t + 1 = 0 \quad \text{or} \quad t - 4 = 0 \]
\[ t = -1 \quad \text{or} \quad t = 4 \]
4. **Interpret the solutions**:
- Discard \( t = -1 \) because time cannot be negative.
- The valid solution is \( t = 4 \), indicating that the balloon hits the ground 4 seconds after being launched.
#### Verification
Check this result by substituting \( t = 4 \) into the original height equation:
\[ h = -16(4)^2 + 48(4) + 64 \]
\[ h = -16(16) + 192 + 64 \]
\[ h = -256 + 192 + 64 \]
\[ h =](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc60d0879-bf1d-4600-af9d-cdd728ad041f%2F865a4c3c-cc53-47fe-a13c-bbb2791575ab%2Fqt3iv7x_processed.png&w=3840&q=75)
Transcribed Image Text:### College Pranks: Calculating the Time for a Water Balloon to Hit the Ground
A student uses rubber tubing to launch a water balloon from the roof of his dormitory. The height \( h \) (in feet) of the balloon, \( t \) seconds after being launched, is given by the formula
\[ h = -16t^2 + 48t + 64. \]
After how many seconds will the balloon hit the ground?
To find out when the water balloon hits the ground, we need to determine when the height \( h \) will be 0 feet. So, we set \( h \) equal to 0 and solve the quadratic equation for \( t \).
#### Step-by-Step Solution
1. **Set the height to 0** (because the balloon hits the ground when its height is 0):
\[ h = -16t^2 + 48t + 64 \]
\[ 0 = -16t^2 + 48t + 64 \]
2. **Factor the quadratic equation**:
\[ 0 = -16(t^2 - 3t - 4) \]
\[ 0 = -16(t + 1)(t - 4) \]
- Factor out the opposite of the Greatest Common Factor (GCF), which is -16.
- Factor the trinomial \( t^2 - 3t - 4 \).
3. **Solve for \( t \)** by setting each factor that contains a variable to 0:
\[ t + 1 = 0 \quad \text{or} \quad t - 4 = 0 \]
\[ t = -1 \quad \text{or} \quad t = 4 \]
4. **Interpret the solutions**:
- Discard \( t = -1 \) because time cannot be negative.
- The valid solution is \( t = 4 \), indicating that the balloon hits the ground 4 seconds after being launched.
#### Verification
Check this result by substituting \( t = 4 \) into the original height equation:
\[ h = -16(4)^2 + 48(4) + 64 \]
\[ h = -16(16) + 192 + 64 \]
\[ h = -256 + 192 + 64 \]
\[ h =
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 1 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, algebra and related others by exploring similar questions and additional content below.Recommended textbooks for you
Algebra and Trigonometry (6th Edition)
Algebra
ISBN:
9780134463216
Author:
Robert F. Blitzer
Publisher:
PEARSON
Contemporary Abstract Algebra
Algebra
ISBN:
9781305657960
Author:
Joseph Gallian
Publisher:
Cengage Learning
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Algebra and Trigonometry (6th Edition)
Algebra
ISBN:
9780134463216
Author:
Robert F. Blitzer
Publisher:
PEARSON
Contemporary Abstract Algebra
Algebra
ISBN:
9781305657960
Author:
Joseph Gallian
Publisher:
Cengage Learning
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Algebra And Trigonometry (11th Edition)
Algebra
ISBN:
9780135163078
Author:
Michael Sullivan
Publisher:
PEARSON
Introduction to Linear Algebra, Fifth Edition
Algebra
ISBN:
9780980232776
Author:
Gilbert Strang
Publisher:
Wellesley-Cambridge Press
College Algebra (Collegiate Math)
Algebra
ISBN:
9780077836344
Author:
Julie Miller, Donna Gerken
Publisher:
McGraw-Hill Education