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The acceleration of an object (in m/s²) is given by the function a(t) = 6 sin(t). The initial velocity of the object is v(0) = - 11 m/s. Round your answers to four decimal places. a) Find an equation v(t) for the object velocity. v(t) = b) Find the object's displacement (in meters) from time 0 to time 3. meters Find the total distance traveled by the object from time 0 to time 3. meters

The acceleration of an object (in m/s²) is given by the function a(t) = 6 sin(t). The initial velocity of the object is v(0) = - 11 m/s. Round your answers to four decimal places. a) Find an equation v(t) for the object velocity. v(t) = b) Find the object's displacement (in meters) from time 0 to time 3. meters Find the total distance traveled by the object from time 0 to time 3. meters

Calculus: Early Transcendentals
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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Question
The acceleration of an object (in m/s²) is given by the function \( a(t) = 6 \sin(t) \). The initial velocity of the object is \( v(0) = -11 \, \text{m/s} \). Round your answers to four decimal places. a) Find an equation \( v(t) \) for the object velocity. \[ v(t) = \] b) Find the object's displacement (in meters) from time 0 to time 3. \[ \] meters c) Find the total distance traveled by the object from time 0 to time 3. \[ \] meters
Transcribed Image Text:The acceleration of an object (in m/s²) is given by the function \( a(t) = 6 \sin(t) \). The initial velocity of the object is \( v(0) = -11 \, \text{m/s} \). Round your answers to four decimal places. a) Find an equation \( v(t) \) for the object velocity. \[ v(t) = \] b) Find the object's displacement (in meters) from time 0 to time 3. \[ \] meters c) Find the total distance traveled by the object from time 0 to time 3. \[ \] meters
**Problem Statement** The traffic flow rate (cars per hour) across an intersection is given by the function \( r(t) = 500 + 900t - 90t^2 \), where \( t \) is measured in hours, and \( t=0 \) corresponds to 6 am. Determine how many cars pass through the intersection between 6 am and 10 am. **Solution Explanation** 1. **Understanding the Problem:** - We need to find the total number of cars passing through the intersection over a specific time period (from 6 am to 10 am). - The flow rate function, \( r(t) \), describes the rate at which cars are passing in cars per hour. 2. **Integration:** - To find the total number of cars passing through the intersection in the given time frame, integrate the flow rate function \( r(t) \) from \( t = 0 \) (6 am) to \( t = 4 \) (10 am). - The integral of \( r(t) \) over this interval will give the total number of cars. 3. **Definite Integration:** - The integration of \( r(t) = 500 + 900t - 90t^2 \) with respect to \( t \) is required. - Find \(\int_{0}^{4} (500 + 900t - 90t^2) \, dt \). 4. **Computing the Integral:** - \(\int (500 + 900t - 90t^2) \, dt = 500t + \frac{900}{2}t^2 - \frac{90}{3}t^3 + C\) - Evaluate from \( t = 0 \) to \( t = 4 \). 5. **Substitute and Calculate:** - Substitute the limits into the integrated function and calculate: - At \( t = 4 \): \( 500(4) + 450(4^2) - 30(4^3) \) - At \( t = 0 \): \( 500(0) + 450(0^2) - 30(0^3) = 0 \) - Compute the difference to find the total number of cars. By working through these steps, you can determine the total number of cars passing
Transcribed Image Text:**Problem Statement** The traffic flow rate (cars per hour) across an intersection is given by the function \( r(t) = 500 + 900t - 90t^2 \), where \( t \) is measured in hours, and \( t=0 \) corresponds to 6 am. Determine how many cars pass through the intersection between 6 am and 10 am. **Solution Explanation** 1. **Understanding the Problem:** - We need to find the total number of cars passing through the intersection over a specific time period (from 6 am to 10 am). - The flow rate function, \( r(t) \), describes the rate at which cars are passing in cars per hour. 2. **Integration:** - To find the total number of cars passing through the intersection in the given time frame, integrate the flow rate function \( r(t) \) from \( t = 0 \) (6 am) to \( t = 4 \) (10 am). - The integral of \( r(t) \) over this interval will give the total number of cars. 3. **Definite Integration:** - The integration of \( r(t) = 500 + 900t - 90t^2 \) with respect to \( t \) is required. - Find \(\int_{0}^{4} (500 + 900t - 90t^2) \, dt \). 4. **Computing the Integral:** - \(\int (500 + 900t - 90t^2) \, dt = 500t + \frac{900}{2}t^2 - \frac{90}{3}t^3 + C\) - Evaluate from \( t = 0 \) to \( t = 4 \). 5. **Substitute and Calculate:** - Substitute the limits into the integrated function and calculate: - At \( t = 4 \): \( 500(4) + 450(4^2) - 30(4^3) \) - At \( t = 0 \): \( 500(0) + 450(0^2) - 30(0^3) = 0 \) - Compute the difference to find the total number of cars. By working through these steps, you can determine the total number of cars passing
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