The height (feet) of an object launched in the air t seconds after launch is given by s= - 16t² +240t+7. Find the time when the ball reaches its maximum height..
The height (feet) of an object launched in the air t seconds after launch is given by s= - 16t² +240t+7. Find the time when the ball reaches its maximum height..
Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018
18th Edition
ISBN:9780079039897
Author:Carter
Publisher:Carter
Chapter9: Quadratic Functions And Equations
Section9.4: Solving Quadratic Equations By Factoring
Problem 59HP
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![### Determining the Time to Reach Maximum Height for a Projectile
#### Problem Statement
The height (feet) of an object launched in the air \( t \) seconds after launch is given by the equation:
\[ s = -16t^2 + 240t + 7 \]
#### Task
Find the time when the ball reaches its maximum height.
The maximum height occurs at \( t = \)
\[ \_\_\_\_\_ \] seconds.
(Simplify your answer. Type an integer or a decimal.)
#### Solution Approach
To determine the time when the ball reaches its maximum height, we'll recognize that the given equation is a quadratic equation in the form of \( s = at^2 + bt + c \), where \( a = -16 \), \( b = 240 \), and \( c = 7 \).
The maximum height of a quadratic equation \( ax^2 + bx + c \) occurs at \( t = -\frac{b}{2a} \).
**Step-by-step Solution:**
1. Identify the coefficients: \( a = -16 \) and \( b = 240 \).
2. Use the formula \( t = -\frac{b}{2a} \).
By substituting the values of \( a \) and \( b \):
\[ t = -\frac{240}{2 \times -16} \]
\[ t = -\frac{240}{-32} \]
\[ t = 7.5 \]
Therefore, the maximum height occurs at \( t = 7.5 \) seconds.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd50d07e0-736c-4d40-9888-4496fe84366c%2F6dda3810-95dc-4168-a7fc-3588b97f10dd%2Fx52gsrf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Determining the Time to Reach Maximum Height for a Projectile
#### Problem Statement
The height (feet) of an object launched in the air \( t \) seconds after launch is given by the equation:
\[ s = -16t^2 + 240t + 7 \]
#### Task
Find the time when the ball reaches its maximum height.
The maximum height occurs at \( t = \)
\[ \_\_\_\_\_ \] seconds.
(Simplify your answer. Type an integer or a decimal.)
#### Solution Approach
To determine the time when the ball reaches its maximum height, we'll recognize that the given equation is a quadratic equation in the form of \( s = at^2 + bt + c \), where \( a = -16 \), \( b = 240 \), and \( c = 7 \).
The maximum height of a quadratic equation \( ax^2 + bx + c \) occurs at \( t = -\frac{b}{2a} \).
**Step-by-step Solution:**
1. Identify the coefficients: \( a = -16 \) and \( b = 240 \).
2. Use the formula \( t = -\frac{b}{2a} \).
By substituting the values of \( a \) and \( b \):
\[ t = -\frac{240}{2 \times -16} \]
\[ t = -\frac{240}{-32} \]
\[ t = 7.5 \]
Therefore, the maximum height occurs at \( t = 7.5 \) seconds.
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