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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 6.1.24c
Chapter 6, Problem 6.1.24c

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?

Verified step by step guidance
1
Identify the given velocity function: \(v(t) = 400 - 20t\), where \(t\) is in minutes and \(v(t)\) is in meters per minute.
Find the time \(t\) when the velocity is 250 m/min by setting \(v(t) = 250\) and solving for \(t\): \(400 - 20t = 250\).
Once you find the time \(t\), recall that the distance traveled is the integral of velocity over time, so the distance \(s(t)\) from \(t=0\) to this time is given by \(s(t) = \int_0^t v(\tau) \, d\tau\).
Set up the integral with the velocity function: \(s(t) = \int_0^t (400 - 20\tau) \, d\tau\).
Evaluate the integral to find the distance traveled up to the time when velocity is 250 m/min.

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Key Concepts

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

Velocity and its Relation to Displacement

Velocity is the rate of change of displacement with respect to time. To find the distance traveled, we analyze the velocity function over time, noting that displacement is the integral of velocity. Understanding how velocity changes helps determine the position at any given time.
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Intro To Related Rates

Solving for Time from Velocity

Given a velocity function v(t), finding when the velocity equals a specific value involves solving an equation for t. This step is crucial to identify the exact time at which the cyclist's velocity reaches 250 m/min, which then allows calculation of the distance traveled up to that time.
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Using The Velocity Function

Definite Integration to Find Distance

Distance traveled over a time interval is found by integrating the velocity function with respect to time between the initial and final times. This process sums the infinitesimal displacements, providing the total distance covered when the velocity reaches the specified value.
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Definition of the Definite Integral
Related Practice
Textbook Question

Where do they meet? Kelly started at noon (t=0) riding a bike from Niwot to Berthoud, a distance of 20 km, with velocity v(t) = 15 / (t + 1)² (decreasing because of fatigue). Sandy started at noon (t=0) riding a bike in the opposite direction from Berthoud to Niwot with velocity u(t) = 20 / (t + 1)² (also decreasing because of fatigue). Assume distance is measured in kilometers and time is measured in hours.


c. When do they meet? How far has each person traveled when they meet?

Textbook Question

6–8. Let R be the region bounded by the curves y = 2−√x,y=2, and x=4 in the first quadrant.

Suppose the shell method is used to determine the volume of the solid generated by revolving R about the line x=4.

c. Write an integral for the volume of the solid using the shell method.

Textbook Question

{Use of Tech} Oscillating motion A mass hanging from a spring is set in motion, and its ensuing velocity is given by v(t) = 2π cos πt, for t≥0. Assume the positive direction is upward and s(0)=0. 


c. At what times does the mass reach its low point the first three times? 

Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

c. If a region is revolved about the x-axis, then in principle, it is possible to use the disk/washer method and integrate with respect to x or to use the shell method and integrate with respect to y.

Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.


Arc length may be negative if f(x) < 0 on part of the interval in question.

Textbook Question

Distance traveled and displacement Suppose an object moves along a line with velocity (in ft/s) v(t)=6−2t, for 0≤t≤6, where t is measured in seconds.


c. Find the distance traveled by the object on the interval 0≤t≤6.