Textbook Question
{Use of Tech} Decreasing velocity A projectile is fired upward, and its velocity in m/s is given by v(t) = 200e^−t/10, for t≥0.
a. Graph the velocity function, for t≥0.
Ch. 6 - Applications of Integration
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 6, Problem 6.RE.9a
Fuel consumption A small plane in flight consumes fuel at a rate (in gal/min) given by
R'(t) ={ 4t^{1/3} if 0 ≤ t ≤ 8 (take-off)
2 if t> 0 (cruising)
a. Find a function R that gives the total fuel consumed, for 0≤t≤8.
Verified step by step guidance1
Identify the given rate of fuel consumption function for the time interval 0 \(\leq\) t \(\leq\) 8, which is R'(t) = 4t^{1/3}. This represents the rate of fuel consumption in gallons per minute during take-off.
Recall that to find the total fuel consumed function R(t), you need to integrate the rate function R'(t) with respect to time t over the interval from 0 to t.
Set up the integral: R(t) = \(\int\) 4t^{1/3} \, dt. This integral will give the total fuel consumed from time 0 up to time t during take-off.
Perform the integration by applying the power rule for integrals: \(\int\) t^{n} \, dt = \(\frac{t^{n+1}\)}{n+1} + C. Here, n = \(\frac{1}{3}\), so integrate accordingly and include the constant of integration C.
Use the initial condition R(0) = 0 (since no fuel is consumed at time zero) to solve for the constant C, ensuring the total fuel consumed function R(t) correctly models the situation.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Rate of Change and Derivatives
The rate of change represents how a quantity changes over time, often expressed as a derivative. In this problem, R'(t) is the rate of fuel consumption, showing how many gallons are used per minute at time t. Understanding derivatives helps interpret and work with rates in real-world contexts.
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Integration to Find Accumulated Quantity
Integration is the reverse process of differentiation and is used to find the total accumulated amount from a rate function. Here, integrating R'(t) over time gives the total fuel consumed, R(t), between 0 and 8 minutes. This concept connects rates to total quantities.
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Piecewise Functions
Piecewise functions define different expressions over different intervals. The fuel consumption rate R'(t) changes form at t=8, requiring careful handling of each interval separately. Understanding piecewise functions ensures correct application of integration and interpretation of the problem.
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Related Practice
Textbook Question
2–3. Displacement, distance, and position Consider an object moving along a line with the following velocities and initial positions. Assume time t is measured in seconds and velocities have units of m/s.
d. Determine the position function s(t) using the Fundamental Theorem of Calculus (Theorem 6.1). Check your answer by finding the position function using the antiderivative method.
v(t) = 12t²-30t+12, for 0 ≤ t ≤ 3; s(0)=1
Textbook Question
Area and volume The region R is bounded by the curves x = y²+2,y=x−4, and y=0 (see figure).
a. Write a single integral that gives the area of R.
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Textbook Question
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
c. If water flows into a tank at a constant rate (for example, 6 gal/min), the volume of water in the tank increases according to a linear function of time.
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.
f. Graph ymax as a function of v0. What is the maximum height when v0=500 m/s,1500 m/s, and 5 km/s?
Textbook Question
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
b. Given only the velocity of an object moving on a line, it is possible to find its displacement, but not its position.