We are standing on the top of a 384 feet tall building and launch a small object upward. The object's vertical position, measured in feet, after t seconds is h(t) 16t² + 32t + 384. What is the highest point that the object reaches? = - feet
We are standing on the top of a 384 feet tall building and launch a small object upward. The object's vertical position, measured in feet, after t seconds is h(t) 16t² + 32t + 384. What is the highest point that the object reaches? = - feet
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Determining the Maximum Height of a Launched Object
We are standing on the top of a 384 feet tall building and launch a small object upward. The object's vertical position, measured in feet, after \( t \) seconds is given by the function:
\[
h(t) = -16t^2 + 32t + 384
\]
What is the highest point that the object reaches?
To find the maximum height, we can determine the vertex of this parabolic function, which is provided by the formula \( h(t) \). The vertex form of a parabola \( h(t) = at^2 + bt + c \) gives the time \( t \) at which the object reaches its maximum height as:
\[
t = -\frac{b}{2a}
\]
Substituting the coefficients \( a = -16 \) and \( b = 32 \) from the given function:
\[
t = -\frac{32}{2(-16)} = \frac{32}{32} = 1
\]
So, the object reaches its highest point at \( t = 1 \) second. Substitute \( t = 1 \) back into the height function to find the maximum height:
\[
h(1) = -16(1)^2 + 32(1) + 384
\]
\[
h(1) = -16 + 32 + 384
\]
\[
h(1) = 400
\]
Thus, the highest point that the object reaches is
\[
\boxed{400} \text{ feet}
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F04ddd72f-30f0-437a-bf8b-c24526128137%2Fc68bac9e-c749-484a-be29-3326b591c52c%2Fze5p8v7_processed.png&w=3840&q=75)
Transcribed Image Text:### Determining the Maximum Height of a Launched Object
We are standing on the top of a 384 feet tall building and launch a small object upward. The object's vertical position, measured in feet, after \( t \) seconds is given by the function:
\[
h(t) = -16t^2 + 32t + 384
\]
What is the highest point that the object reaches?
To find the maximum height, we can determine the vertex of this parabolic function, which is provided by the formula \( h(t) \). The vertex form of a parabola \( h(t) = at^2 + bt + c \) gives the time \( t \) at which the object reaches its maximum height as:
\[
t = -\frac{b}{2a}
\]
Substituting the coefficients \( a = -16 \) and \( b = 32 \) from the given function:
\[
t = -\frac{32}{2(-16)} = \frac{32}{32} = 1
\]
So, the object reaches its highest point at \( t = 1 \) second. Substitute \( t = 1 \) back into the height function to find the maximum height:
\[
h(1) = -16(1)^2 + 32(1) + 384
\]
\[
h(1) = -16 + 32 + 384
\]
\[
h(1) = 400
\]
Thus, the highest point that the object reaches is
\[
\boxed{400} \text{ feet}
\]
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