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Ch. 6 - Applications of Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 6 - Applications of IntegrationProblem 6.1.25c
Chapter 6, Problem 6.1.25c

Flying into a headwind The velocity (in mi/hr) of an airplane flying into a headwind is given by v(t) = 30(16−t²), for 0≤t≤3. Assume s(0)=0 and t is measured in hours.


c. How far has the airplane traveled at the instant its velocity reaches 400 mi/hr?

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1
First, understand that the velocity function is given by \(v(t) = 30(16 - t^{2})\), where \(t\) is time in hours and \(v(t)\) is velocity in miles per hour. We want to find the distance traveled when the velocity reaches 400 mi/hr.
Set the velocity equal to 400 and solve for \(t\): \(30(16 - t^{2}) = 400\). This will give the time(s) at which the velocity is 400 mi/hr.
Solve the equation for \(t\): divide both sides by 30 to isolate the quadratic term, then rearrange to form a quadratic equation and solve for \(t\).
Once you find the appropriate value of \(t\) within the interval \(0 \leq t \leq 3\), use the position function \(s(t)\) to find the distance traveled. Since velocity is the derivative of position, \(s(t) = \int v(t) \, dt + s(0)\).
Integrate \(v(t) = 30(16 - t^{2})\) with respect to \(t\) to find \(s(t)\), then substitute the value of \(t\) found earlier into \(s(t)\) to find the distance traveled at that instant.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Velocity and Position Relationship

Velocity is the rate of change of position with respect to time. To find the distance traveled, you integrate the velocity function over the given time interval. Here, s(t) represents the position, and its derivative s'(t) equals the velocity v(t).
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Derivatives Applied To Velocity

Solving for Time from Velocity

To find when the velocity reaches a specific value, set the velocity function equal to that value and solve for time t. This step is crucial to determine the exact instant at which the airplane's velocity is 400 mi/hr.
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Using The Velocity Function

Definite Integration for Displacement

Once the time t is found, calculate the distance traveled by integrating the velocity function from the initial time to t. This definite integral gives the total displacement, which is the airplane's traveled distance at that instant.
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Definition of the Definite Integral
Related Practice
Textbook Question

Bike race Theo and Sasha start at the same place on a straight road, riding bikes with the following velocities (measured in mi/hr). Assume t is measured in hours.

Theo: vT(t)=10, for t≥0

Sasha: vS(t)=15t, for 0≤t≤1, and vS(t)=15, for t>1


c. If the riders ride for 2 hr, who rides farther? Interpret your answer geometrically using the graphs of part (a). 

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Textbook Question

9–10. Velocity graphs The figures show velocity functions for motion along a line. Assume the motion begins with an initial position of s(0)=0. Determine the following.

d. A piecewise function for s(t)

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Textbook Question

Cycling distance A cyclist rides down a long straight road with a velocity (in m/min) given by v(t) = 400−20t, for 0≤t≤10, where t is measured in minutes.


c. How far has the cyclist traveled when her velocity is 250 m/min?

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Textbook Question

9–10. Velocity graphs The figures show velocity functions for motion along a line. Assume the motion begins with an initial position of s(0)=0. Determine the following.

d. A piecewise function for s(t)

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Textbook Question

Day hike The velocity (in mi/hr) of a hiker walking along a straight trail is given by v(t) = 3 sin² πt/2, for 0≤t≤4. Assume s(0)=0 and t is measured in hours. 


c. What is the hiker’s position at t=3?

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Textbook Question

13–16. Displacement from velocity Consider an object moving along a line with the given velocity v. Assume time t is measured in seconds and velocities have units of m/s.


c. Find the distance traveled over the given interval.


v(t) = 3t²−6t on [0, 3]

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