### Problem StatementThe 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 Steps1. **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. ### SummaryAfter 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.

For an equation for distance, with respect to time, we can find the velocity and acceleration, by…

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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