Textbook Question
55–58. Marginal cost Consider the following marginal cost functions.
a. Find the additional cost incurred in dollars when production is increased from 100 units to 150 units.
C′(x) = 300+10x−0.01x²
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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 6.7.42a
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).
a. How much work is required to pump the water out of the trough (to the level of the top of the trough) when it is full?
Verified step by step guidance1
Identify the shape and dimensions of the trough: it has a semicircular cross section with radius \(r = 0.25\) m and length \(L = 3\) m.
Set up a coordinate system for the vertical direction, letting \(y\) represent the depth of a thin horizontal slice of water measured from the bottom of the trough (where \(y=0\)) up to the top (where \(y=0.25\) m).
Express the width of the water slice at height \(y\). Since the cross section is semicircular, the width \(w(y)\) is twice the horizontal distance from the center to the edge, which can be found using the circle equation: \(w(y) = 2 \sqrt{r^2 - (r - y)^2}\).
Calculate the volume of a thin slice of water at height \(y\) with thickness \(dy\): \(dV = w(y) \times L \times dy\).
Determine the work to pump this slice to the top of the trough. The distance the water must be lifted is \((r - y)\), and the weight density of water is \(\rho g\) (where \(\rho\) is the density of water and \(g\) is acceleration due to gravity). The work for the slice is \(dW = \rho g \times dV \times (r - y)\). Integrate \(dW\) from \(y=0\) to \(y=r\) to find the total 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 pumping water, the force varies with the weight of each water slice and the distance it must be moved. Calculating work involves integrating the product of force and distance over the volume of water.
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Volume and Cross-Sectional Area of a Semicircle
The trough has a semicircular cross section, so the area of a horizontal slice depends on the radius and the height of the water at that slice. Understanding how to express the area of a semicircle segment as a function of depth is essential for setting up the integral.
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Density and Weight of Water
The force exerted by the water is its weight, which depends on the volume, density, and gravitational acceleration. Knowing the density of water allows conversion from volume to weight, which is necessary to calculate the force in the work integral.
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Related Practice
Textbook Question
Given the velocity function of an object moving along a line, explain how definite integrals can be used to find the displacement of the object.
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Textbook Question
6–8. Let R be the region bounded by the curves y = 2−√x,y=2, and x=4 in the first quadrant.
Suppose the shell method is used to determine the volume of the solid generated by revolving R about the line x=4.
a. What is the radius of a cylindrical shell at a point x in [0, 4]?
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Textbook Question
{Use of Tech} Oscillating motion A mass hanging from a spring is set in motion, and its ensuing velocity is given by v(t) = 2π cos πt, for t≥0. Assume the positive direction is upward and s(0)=0.
a. Determine the position function, for t≥0.
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Textbook Question
13–16. Displacement from velocity Consider an object moving along a line with the given velocity v. Assume time t is measured in seconds and velocities have units of m/s.
b. Find the displacement over the given interval.
v(t) = 50e^−2t on [0, 4]
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Textbook Question
A vertical spring A 10-kg mass is attached to a spring that hangs vertically and is stretched 2 m from the equilibrium position of the spring. Assume a linear spring with F(x) = kx.
a. How much work is required to compress the spring and lift the mass 0.5 m?
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