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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 6.1.37d
Chapter 6, Problem 6.1.37d

Acceleration A drag racer accelerates at a(t)=88 ft/s². Assume v(0)=0, s(0)=0, and t is measured in seconds.


d. How long does it take the racer to travel 300 ft?

Verified step by step guidance
1
Identify the given acceleration function: \(a(t) = 88\) ft/s², which is constant.
Since acceleration is the derivative of velocity, integrate \(a(t)\) with respect to \(t\) to find the velocity function: \(v(t) = \int a(t) \, dt = \int 88 \, dt\).
Use the initial condition \(v(0) = 0\) to solve for the constant of integration in the velocity function.
Next, integrate the velocity function \(v(t)\) with respect to \(t\) to find the position function \(s(t)\): \(s(t) = \int v(t) \, dt\).
Use the initial condition \(s(0) = 0\) to solve for the constant of integration in the position function, then set \(s(t) = 300\) ft and solve for \(t\) to find the time it takes to travel 300 ft.

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Key Concepts

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

Acceleration and Its Relationship to Velocity and Position

Acceleration is the rate of change of velocity with respect to time. Given acceleration a(t), velocity v(t) can be found by integrating a(t). Similarly, position s(t) is found by integrating velocity. Understanding these relationships allows us to move from acceleration to position over time.
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Derivatives Applied To Acceleration

Initial Conditions in Integration

Initial conditions such as v(0) = 0 and s(0) = 0 provide specific values needed to solve the constants of integration when finding velocity and position functions. These conditions ensure the solution matches the physical scenario described.
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Initial Value Problems

Solving for Time from Position Function

Once the position function s(t) is determined, solving for the time t when s(t) equals a given distance (300 ft) involves algebraic manipulation. This step finds the exact time required for the racer to travel the specified distance.
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Related Practice
Textbook Question

Use the region R that is bounded by the graphs of y=1+√x,x=4, and y=1 complete the exercises.


Region R is revolved about the y-axis to form a solid of revolution whose cross sections are washers.


d. Write an integral for the volume of the solid.

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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

Use the region R that is bounded by the graphs of y=1+√x,x=4, and y=1 complete the exercises.


Region R is revolved about the x-axis to form a solid of revolution whose cross sections are washers.


d. Write an integral for the volume of the solid.

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Textbook Question

Displacement and distance from velocity Consider the velocity function shown below of an object moving along a line. Assume time is measured in seconds and distance is measured in meters. The areas of four regions bounded by the velocity curve and the t-axis are also given.

d. What is the displacement of the object over the interval [0, 8]? 

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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

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


d. Let f(x)=12x^2.. The area of the surface generated when the graph of f on [−4, 4] is revolved about the y-axis is twice the area of the surface generated when the graph of f on [0, 4] is revolved about the y-axis.

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