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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 6.7.42b
Chapter 6, Problem 6.7.42b

Emptying a water trough A water trough has a semicircular cross section with a radius of 0.25 m and a length of 3 m (see figure).
b. If the length is doubled, is the required work doubled? Explain.

Verified step by step guidance
1
Step 1: Understand the problem context. The trough has a semicircular cross section with radius 0.25 m and length 3 m. The work required to empty the trough depends on the volume of water and the distance it must be lifted.
Step 2: Recall that work is calculated as the integral of force times distance. Here, force is related to the weight of the water, which depends on the volume of water and the density of water.
Step 3: The volume of water in the trough is the area of the semicircular cross section multiplied by the length. The area of a semicircle is given by \(\frac{1}{2} \pi r^2\), so the volume is \(V = \frac{1}{2} \pi (0.25)^2 \times 3\).
Step 4: If the length is doubled, the volume of water doubles because the cross-sectional area remains the same but the length doubles. Therefore, the total weight of the water doubles.
Step 5: Since the work is the integral of weight times the distance the water is lifted, and the lifting distance does not change with length, doubling the length doubles the volume and weight, thus doubling the required work.

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Key Concepts

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

Work Done by a Variable Force

Work is the integral of force over distance. When emptying a trough, the force varies with the depth of water, so the work is calculated by integrating the weight of water lifted at each depth multiplied by the distance it is lifted.
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Work Done On A Spring (Hooke's Law)

Volume and Cross-Sectional Area

The volume of water in the trough depends on the cross-sectional area and the length. Doubling the length doubles the volume, which directly affects the total weight and thus the work required to empty the trough.
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Introduction to Cross Sections

Linear Scaling of Work with Length

Since the cross-sectional shape and height remain constant, the work required to empty the trough scales linearly with its length. Doubling the length doubles the volume and weight of water, so the total work required is also doubled.
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Introduction To Work
Related Practice
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? 

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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 a region is revolved about the x-axis, then in principle, it is possible to use the disk/washer method and integrate with respect to x or to use the shell method and integrate with respect to y.

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Textbook Question

A nonlinear spring Hooke’s law is applicable to idealized (linear) springs that are not stretched or compressed too far from their equilibrium positions. Consider a nonlinear spring whose restoring force is given by F(x) = 16x−0.1x³, for |x|≤7. 

b. How much work is done in stretching the spring from its equilibrium position (x=0) to x=1.5?

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Textbook Question

Volumes without calculus Solve the following problems with and without calculus. A good picture helps.


b. A cube is inscribed in a right circular cone with a radius of 1 and a height of 3. What is the volume of the cube?

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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.


c. When will the tank be full?

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Textbook Question

Emptying a water trough A water trough has a semicircular cross section with a radius of 0.25 m and a length of 3 m (see figure).

c. If the radius is doubled, is the required work doubled? Explain.

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