The height (in meters) of a ballistic t seconds after being launched is modeled by the function h (t) = 60 + 80t - 5t2. Find the height, vertical velocity, and acceleration of the object after 10 seconds. Include units. If any of the numbers are negative, explain what that means in the real-world application.
The height (in meters) of a ballistic t seconds after being launched is modeled by the function h (t) = 60 + 80t - 5t2. Find the height, vertical velocity, and acceleration of the object after 10 seconds. Include units. If any of the numbers are negative, explain what that means in the real-world application.
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...
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![### Problem Statement
The height (in meters) of a ballistic \( t \) seconds after being launched is modeled by the function:
\[ h(t) = 60 + 80t - 5t^2 \]
Find the height, vertical velocity, and acceleration of the object after 10 seconds. Include units. If any of the numbers are negative, explain what that means in the real-world application.
### Solution Steps
1. **Height Calculation**:
To find the height of the object at \( t = 10 \) seconds, substitute \( t \) with 10 in the height function \( h(t) \).
\[
h(10) = 60 + 80(10) - 5(10)^2
= 60 + 800 - 500
= 360 \text{ meters}
\]
2. **Vertical Velocity Calculation**:
The vertical velocity can be found by taking the derivative of the height function \( h(t) \) with respect to time \( t \).
\[
v(t) = \frac{dh(t)}{dt} = 80 - 10t
\]
Now, substitute \( t \) with 10:
\[
v(10) = 80 - 10(10)
= 80 - 100
= -20 \text{ meters per second}
\]
3. **Acceleration Calculation**:
The acceleration is the derivative of the velocity function (which corresponds to the second derivative of the height function).
\[
a(t) = \frac{dv(t)}{dt} = -10 \text{ meters per second squared}
\]
### Interpretation of Negative Values
- The **negative velocity** of \(-20\) meters per second at \( t = 10 \) seconds indicates that the object is moving downward at that instant.
- The **constant negative acceleration** of \(-10\) meters per second squared represents the effect of gravity acting on the object, causing it to decelerate while going up and accelerate while coming down.
### Summary
After 10 seconds:
- The height of the object is 360 meters.
- The vertical velocity of the object is \(-20\) meters per second (moving downward).
- The acceleration due to gravity is \(-10\) meters per second squared.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa3d3ac4a-874a-47f2-996d-ede45746d6c1%2F8b7c6a3e-834b-4a17-8a8e-e69521477db5%2Fgqhfzxf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement
The height (in meters) of a ballistic \( t \) seconds after being launched is modeled by the function:
\[ h(t) = 60 + 80t - 5t^2 \]
Find the height, vertical velocity, and acceleration of the object after 10 seconds. Include units. If any of the numbers are negative, explain what that means in the real-world application.
### Solution Steps
1. **Height Calculation**:
To find the height of the object at \( t = 10 \) seconds, substitute \( t \) with 10 in the height function \( h(t) \).
\[
h(10) = 60 + 80(10) - 5(10)^2
= 60 + 800 - 500
= 360 \text{ meters}
\]
2. **Vertical Velocity Calculation**:
The vertical velocity can be found by taking the derivative of the height function \( h(t) \) with respect to time \( t \).
\[
v(t) = \frac{dh(t)}{dt} = 80 - 10t
\]
Now, substitute \( t \) with 10:
\[
v(10) = 80 - 10(10)
= 80 - 100
= -20 \text{ meters per second}
\]
3. **Acceleration Calculation**:
The acceleration is the derivative of the velocity function (which corresponds to the second derivative of the height function).
\[
a(t) = \frac{dv(t)}{dt} = -10 \text{ meters per second squared}
\]
### Interpretation of Negative Values
- The **negative velocity** of \(-20\) meters per second at \( t = 10 \) seconds indicates that the object is moving downward at that instant.
- The **constant negative acceleration** of \(-10\) meters per second squared represents the effect of gravity acting on the object, causing it to decelerate while going up and accelerate while coming down.
### Summary
After 10 seconds:
- The height of the object is 360 meters.
- The vertical velocity of the object is \(-20\) meters per second (moving downward).
- The acceleration due to gravity is \(-10\) meters per second squared.
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