Suppose that a baseball is thrown upward with an initial velocity of 44 feet per second (30 miles per hour) and it is released 6 feet above the ground. Its height h after t seconds is given by h = - 16t² +44t+ 6. After how many seconds does the baseball reach a maximum height? Estimate this height. The baseball reaches the maximum height after approximately (Round to the nearest integer as needed.) The estimated maximum height is ft. (Round to the nearest integer as needed.) sec from the time its being thrown upward.
Suppose that a baseball is thrown upward with an initial velocity of 44 feet per second (30 miles per hour) and it is released 6 feet above the ground. Its height h after t seconds is given by h = - 16t² +44t+ 6. After how many seconds does the baseball reach a maximum height? Estimate this height. The baseball reaches the maximum height after approximately (Round to the nearest integer as needed.) The estimated maximum height is ft. (Round to the nearest integer as needed.) sec from the time its being thrown upward.
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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![### Problem Statement:
Suppose that a baseball is thrown upward with an initial velocity of 44 feet per second (30 miles per hour) and it is released 6 feet above the ground. Its height \( h \) after \( t \) seconds is given by \( h = -16t^2 + 44t + 6 \). After how many seconds does the baseball reach a maximum height? Estimate this height.
### Solution:
**1. Determining the time to reach the maximum height:**
The baseball reaches the maximum height after approximately \[ \]
\[\text{sec from the time it is being thrown upward.}\]
\[\text{(Round to the nearest integer as needed.)}\]
**2. Estimating the maximum height:**
The estimated maximum height is \[ \]
\[\text{ft.}\]
\[\text{(Round to the nearest integer as needed.)}\]
### Explanation:
**Quadratic Equation in Vertex Form:**
The function \( h = -16t^2 + 44t + 6 \) represents a parabola opening downward (since the coefficient of \( t^2 \) is negative). The vertex of this parabola will provide the maximum height.
To find the vertex, use the formula for the time to reach maximum height (t) in a quadratic equation \( at^2 + bt + c \):
\[ t = -\dfrac{b}{2a} \]
where
- \( a = -16 \)
- \( b = 44 \)
Substitute these values into the formula:
\[ t = -\dfrac{44}{2(-16)} = \dfrac{44}{32} = 1.375 \]
So the baseball reaches its maximum height at t ≈ 1.38 seconds.
**Height Calculation:**
Substitute \( t = 1.375 \) back into the height equation to find the maximum height \( h \):
\[ h = -16(1.375)^2 + 44(1.375) + 6 \]
\[ h ≈ -16(1.89) + 60.5 + 6 \]
\[ h ≈ -30.24 + 60.5 + 6 ≈ 36.26 \]
So, the estimated maximum height is approximately 36 feet, when rounded to the nearest integer.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa5da6b72-6f0b-4b5b-b99e-129f8d660be8%2Fb57cd723-dec2-452c-9a6d-7bf0f4abf755%2F9ezvgzh_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Statement:
Suppose that a baseball is thrown upward with an initial velocity of 44 feet per second (30 miles per hour) and it is released 6 feet above the ground. Its height \( h \) after \( t \) seconds is given by \( h = -16t^2 + 44t + 6 \). After how many seconds does the baseball reach a maximum height? Estimate this height.
### Solution:
**1. Determining the time to reach the maximum height:**
The baseball reaches the maximum height after approximately \[ \]
\[\text{sec from the time it is being thrown upward.}\]
\[\text{(Round to the nearest integer as needed.)}\]
**2. Estimating the maximum height:**
The estimated maximum height is \[ \]
\[\text{ft.}\]
\[\text{(Round to the nearest integer as needed.)}\]
### Explanation:
**Quadratic Equation in Vertex Form:**
The function \( h = -16t^2 + 44t + 6 \) represents a parabola opening downward (since the coefficient of \( t^2 \) is negative). The vertex of this parabola will provide the maximum height.
To find the vertex, use the formula for the time to reach maximum height (t) in a quadratic equation \( at^2 + bt + c \):
\[ t = -\dfrac{b}{2a} \]
where
- \( a = -16 \)
- \( b = 44 \)
Substitute these values into the formula:
\[ t = -\dfrac{44}{2(-16)} = \dfrac{44}{32} = 1.375 \]
So the baseball reaches its maximum height at t ≈ 1.38 seconds.
**Height Calculation:**
Substitute \( t = 1.375 \) back into the height equation to find the maximum height \( h \):
\[ h = -16(1.375)^2 + 44(1.375) + 6 \]
\[ h ≈ -16(1.89) + 60.5 + 6 \]
\[ h ≈ -30.24 + 60.5 + 6 ≈ 36.26 \]
So, the estimated maximum height is approximately 36 feet, when rounded to the nearest integer.
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