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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 6.1.24b
Chapter 6, Problem 6.1.24b

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.


b. How far does the cyclist travel in the first 10 min?

Verified step by step guidance
1
Identify the velocity function given: \(v(t) = 400 - 20t\), where \(t\) is in minutes and \(v(t)\) is in meters per minute.
Recall that the distance traveled over a time interval can be found by integrating the velocity function over that interval. So, the distance \(D\) traveled from \(t=0\) to \(t=10\) is given by the definite integral \(D = \int_0^{10} v(t) \, dt\).
Set up the integral explicitly: \(D = \int_0^{10} (400 - 20t) \, dt\).
Integrate the function term-by-term: the integral of \(400\) with respect to \(t\) is \$400t\(, and the integral of \)-20t\( with respect to \)t\( is \)-10t^2$.
Evaluate the resulting expression \(400t - 10t^2\) at the upper limit \(t=10\) and subtract the value at the lower limit \(t=0\) to find the total distance traveled.

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Key Concepts

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

Velocity as a Function of Time

Velocity describes the rate of change of position with respect to time. In this problem, velocity is given as a function v(t) = 400 - 20t, which means the cyclist's speed changes linearly over time from 400 m/min to 200 m/min during the first 10 minutes.
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Using The Velocity Function

Definite Integral for Displacement

The total distance traveled over a time interval can be found by integrating the velocity function over that interval. The definite integral of v(t) from t=0 to t=10 gives the net displacement, representing how far the cyclist has traveled in those 10 minutes.
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Definition of the Definite Integral

Limits of Integration and Time Interval

The problem specifies the time interval 0 ≤ t ≤ 10 minutes. Setting the correct limits of integration ensures the calculation covers the entire duration of interest, capturing the cyclist's motion from start to 10 minutes exactly.
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Improper Integrals: Infinite Intervals
Related Practice
Textbook Question

Variable gravity At Earth’s surface, the acceleration due to gravity is approximately g=9.8 m/s² (with local variations). However, the acceleration decreases with distance from the surface according to Newton’s law of gravitation. At a distance of y meters from Earth’s surface, the acceleration is given by a(y) = - g / (1+y/R)², where R=6.4×10⁶ m is the radius of Earth.


b. Use the Chain Rule to show that dv/dt = 1/2 d/dy(v²).

Textbook Question

Power and energy The terms power and energy are often used interchangeably, but they are quite different. Energy is what makes matter move or heat up and is measured in units of joules (J) or Calories (Cal), where 1 Cal=4184 J. One hour of walking consumes roughly 10⁶ J, or 250 Cal. On the other hand, power is the rate at which energy is used and is measured in watts (W; 1W=1 J/s). Other useful units of power are kilowatts (1 kW=10³ W) and megawatts (1 MW=10⁶ W). If energy is used at a rate of 1 kW for 1 hr, the total amount of energy used is 1 kilowatt-hour (kWh), which is 3.6×10⁶ J. Suppose the power function of a large city over a 24-hr period is given by P(t) = E'(t) = 300 - 200 sin πt/12, where P is measured in megawatts and t=0 corresponds to 6:00 P.M. (see figure).


b. Burning 1 kg of coal produces about 450 kWh of energy. How many kilograms of coal are required to meet the energy needs of the city for 1 day? For 1 year? 

Textbook Question

Displacement and distance from velocity Consider the graph shown in the figure, which gives the velocity of an object moving along a line. Assume time is measured in hours and distance is measured in miles. The areas of three regions bounded by the velocity curve and the t-axis are also given.

b. What is the displacement of the object over the interval [0,3]?

Textbook Question

Filling a tank A 2000-liter cistern is empty when water begins flowing into it (at t=0 at a rate (in L/min) given by Q′(t) = 3√t, where t is measured in minutes.


b. Find the function that gives the amount of water in the tank at any time t≥0.

Textbook Question

"Determine whether the following statements are true and give an explanation or counterexample.


b. The volume of a hemisphere can be computed using the disk method. "

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Textbook Question

21–30. {Use of Tech} Arc length by calculator


b. If necessary, use technology to evaluate or approximate the integral.

y = cos 2x, for 0 ≤ x ≤ π