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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 6.7.42c
Chapter 6, Problem 6.7.42c

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.

Verified step by step guidance
1
Step 1: Understand the problem context. The water trough has a semicircular cross section with radius \(r = 0.25\) m and length \(L = 3\) m. The work required to empty the trough depends on the volume of water and the height it must be lifted.
Step 2: Recall the formula for work done in lifting water. Work is the integral of force times distance. The force is the weight of the water, which depends on the volume and density, and the distance is the height the water is lifted.
Step 3: Express the volume of water in terms of the radius. The cross-sectional area of the semicircle is \(A = \frac{1}{2} \pi r^2\), so the volume is \(V = A \times L = \frac{1}{2} \pi r^2 L\).
Step 4: Analyze how doubling the radius affects the volume and the height the water must be lifted. Doubling the radius changes the cross-sectional area by a factor of \(4\) (since area depends on \(r^2\)), and the height the water must be lifted also changes proportionally to the radius.
Step 5: Conclude that the work required is not simply doubled when the radius is doubled because work depends on both the volume (which scales with \(r^2\)) and the lifting height (which scales with \(r\)), so the total work scales with \(r^3\).

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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 in calculus is often calculated as the integral of a force over a distance. When emptying a trough, the force varies with the depth of the water, so the work is found by integrating the weight of water lifted at each layer times the distance it is lifted.
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Volume and Cross-Sectional Area of a Semicircle

The volume of water in the trough depends on the semicircular cross-sectional area multiplied by the length. The area of a semicircle is (1/2)πr², so doubling the radius increases the area, and thus the volume, by a factor of four.
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Introduction to Cross Sections

Relationship Between Radius and Work Required

Doubling the radius affects both the volume of water and the height it must be lifted. Since volume scales with the square of the radius and the lifting height scales linearly, the total work does not simply double but increases by a larger factor.
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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

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

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.

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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

Distance traveled and displacement Suppose an object moves along a line with velocity (in ft/s) v(t)=6−2t, for 0≤t≤6, where t is measured in seconds.


c. Find the distance traveled by the object on the interval 0≤t≤6.

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