Skip to main content
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 6.1.37c
Chapter 6, Problem 6.1.37c

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


c. At this rate, how long will it take the racer to travel 1/4 mi?

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 \(v(t)\): \[v(t) = \int a(t) \, dt = \int 88 \, dt = 88t + C_1\] Use the initial condition \(v(0) = 0\) to solve for \(C_1\).
Next, velocity is the derivative of position, so integrate \(v(t)\) to find the position function \(s(t)\): \[s(t) = \int v(t) \, dt = \int 88t \, dt = 44t^2 + C_2\] Use the initial condition \(s(0) = 0\) to solve for \(C_2\).
Convert the distance to consistent units: Since \(1\) mile = \(5280\) feet, \(\frac{1}{4}\) mile = \(\frac{1}{4} \times 5280 = 1320\) feet.
Set the position function equal to \(1320\) feet and solve for \(t\): \[44t^2 = 1320\] Solve this equation to find the time \(t\) it takes to travel \(\frac{1}{4}\) mile.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
4m
Was this helpful?

Key Concepts

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

Acceleration and Its Relationship to Velocity and Displacement

Acceleration is the rate of change of velocity with respect to time. Given a constant acceleration, velocity can be found by integrating acceleration over time, and displacement can be found by integrating velocity. This relationship allows us to determine how velocity and position evolve from acceleration.
Recommended video:
08:56
Using The Acceleration Function Example 1

Integration of Constant Acceleration

When acceleration is constant, velocity increases linearly over time, and displacement follows a quadratic function of time. Specifically, velocity is a(t) multiplied by time plus initial velocity, and displacement is (1/2) times acceleration times time squared plus initial displacement.
Recommended video:
08:14
Using The Acceleration Function

Unit Conversion and Distance Measurement

To solve for time when given a distance in miles, it is essential to convert miles to feet to match the units of acceleration (ft/s²). Since 1 mile equals 5280 feet, converting 1/4 mile to feet ensures consistent units for accurate calculation.
Recommended video:
5:10
Evaluate Composite Functions - Values on Unit Circle
Related Practice
Textbook Question

Oscillating growth rates Some species have growth rates that oscillate with an (approximately) constant period P. Consider the growth rate function N'(t) = r+A sin 2πt/P, where A and r are constants with units of individuals/yr, and t is measured in years. A species becomes extinct if its population ever reaches 0 after t=0.


c. Suppose P=10, A=50, and r=5. If the initial population is N(0)=10, does the population ever become extinct? Explain.

42
views
Textbook Question

Region R is revolved about the line y=1 to form a solid of revolution.


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

73
views
Textbook Question

Let R be the region bounded by the curve y=√cos x and the x-axis on [0, π/2]. A solid of revolution is obtained by revolving R about the x-axis (see figures). 


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

64
views
Textbook Question

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

c. The work required to lift a 10-kg object vertically 10 m is the same as the work required to lift a 20-kg object vertically 5 m.

49
views
Textbook Question

Blood flow A typical human heart pumps 70 mL of blood (the stroke volume) with each beat. Assuming a heart rate of 60 beats/min (1 beat/s), a reasonable model for the outflow rate of the heart is V′(t)=70(1+sin 2πt), where V(t) is the amount of blood (in milliliters) pumped over the interval [0,t],V(0)=0 and t is measured in seconds.


c. What is the cardiac output over a period of 1 min? (Use calculus; then check your answer with algebra.)

97
views
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.

c. The position at t=5

40
views