Domanda del libro di testo
Oil production An oil refinery produces oil at a variable rate given by Q'(t) = <1x3 matrix>, where is measured in days and is measured in barrels.
a. How many barrels are produced in the first 35 days?
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Ch. 6 - Applications of Integration
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- 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: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 6, Problema 6.1.48a
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.
a. How much water flows into the cistern in 1 hour?
Guida verificata passo dopo passo1
Identify the given rate of water flow into the cistern as a function of time: \(Q'(t) = 3\sqrt{t}\) liters per minute, where \(t\) is in minutes.
Recognize that \(Q'(t)\) represents the derivative of the volume of water \(Q(t)\) with respect to time, so to find the total volume of water that has flowed in by time \(t\), you need to integrate \(Q'(t)\) over the interval from 0 to \(t\).
Set up the definite integral to find the total volume of water that has flowed into the cistern in 1 hour (which is 60 minutes):
\(\displaystyle Q(60) = \int_0^{60} 3\sqrt{t} \, dt\)
Rewrite the integrand \(3\sqrt{t}\) as \(3t^{1/2}\) to make integration straightforward.
Integrate \(3t^{1/2}\) with respect to \(t\) using the power rule for integration:
\(\int 3t^{1/2} \, dt = 3 \cdot \frac{t^{3/2}}{\frac{3}{2}} + C = 2t^{3/2} + C\).
Then evaluate this antiderivative from 0 to 60 to find the total volume of water that has flowed into the cistern in 1 hour.
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Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.
Definite Integral
A definite integral calculates the total accumulation of a quantity over an interval. In this problem, integrating the rate function Q′(t) from 0 to 60 minutes gives the total volume of water that has flowed into the cistern during that time.
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Percorso guidato
05:43
Definition of the Definite Integral
Rate of Change and Accumulation
The rate function Q′(t) represents how fast water flows into the tank at any time t. Understanding that integrating this rate over time accumulates the total amount of water is essential to solving the problem.
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Percorso guidato
04:16Intro To Related Rates
Units and Time Conversion
Since the rate is given in liters per minute and time is in minutes, it is important to convert the time interval correctly (1 hour = 60 minutes) to ensure the integral limits match the units and yield a meaningful total volume.
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5:10Evaluate Composite Functions - Values on Unit Circle
Pratica correlata
Domanda del libro di testo
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
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Domanda del libro di testo
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.
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Domanda del libro di testo
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.
a. Suppose P=10, A=20, and r=0. If the initial population is N(0)=10, does the population ever become extinct? Explain.
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Domanda del libro di testo
Work in a gravitational field For large distances from the surface of Earth, the gravitational force is given by F(x) = GMm / (x+R)², where G = 6.7×10^−11 N m²/kg² is the gravitational constant, M = 6×10^24 kg is the mass of Earth, m is the mass of the object in the gravitational field, R = 6.378×10⁶ m is the radius of Earth, and x≥0 is the distance above the surface of Earth (in meters).
a. How much work is required to launch a rocket with a mass of 500 kg in a vertical flight path to a height of 2500 km (from Earth’s surface)?
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Domanda del libro di testo
A surface is generated by revolving the line f(x)=2−x, for 0≤x≤2, about the x-axis. Find the area of the resulting surface in the following ways.
a. Using calculus
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