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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 6.7.39b
Chapter 6, Problem 6.7.39b

Emptying a conical tank A water tank is shaped like an inverted cone with height 6 m and base radius 1.5 m (see figure).
b. Is it true that it takes half as much work to pump the water out of the tank when it is filled to half its depth as when it is full? Explain.
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Verified step by step guidance
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Understand that the work required to pump water out of the tank depends on the volume of water and the distance each water element must be lifted.
Set up the coordinate system with the vertex of the cone at the bottom (y=0) and the top of the tank at y=6 m. The radius of the water surface at height y is proportional to y, given by the similarity ratio \(r(y) = \frac{1.5}{6} y = 0.25 y\).
Express the volume of a thin horizontal slice of water at height y with thickness \(dy\) as \(dV = \pi r(y)^2 dy = \pi (0.25 y)^2 dy = \pi \times 0.0625 y^2 dy\).
The work to lift this slice to the top (y=6) is \(dW = \rho g dV (6 - y)\), where \(\rho\) is the density of water and \(g\) is acceleration due to gravity. Integrate \(dW\) from \(y=0\) to \(y=6\) to find the total work when the tank is full.
Repeat the integration from \(y=0\) to \(y=3\) (half the depth) to find the work when the tank is half full. Compare the two results to determine if the work for half the depth is half the work for the full tank.

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Key Concepts

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

Work Done by Pumping Water

Work in this context is the energy required to move water against gravity from the tank to the top. It is calculated by integrating the force (weight of water) times the distance each water layer is lifted. Since the force depends on volume and height, work is not simply proportional to volume.
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Volume of Water in a Conical Tank

The volume of water in a cone depends on the height of the water level. Because the radius changes linearly with height, the volume varies with the cube of the height. This nonlinear relationship affects how much water is present at half the depth compared to full.
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Relationship Between Height and Radius in a Cone

In a cone, the radius at any height is proportional to that height due to similar triangles. For this tank, radius r = (1.5/6) * height. This relationship is essential to express volume and work integrals in terms of height, enabling calculation of work done for different water depths.
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Related Practice
Textbook Question

Different axes of revolution Suppose R is the region bounded by y=f(x) and y=g(x) on the interval [a, b], where f(x)≥g(x).


b. How is this formula changed if x0>b?

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Textbook Question

Emptying a cylindrical tank A cylindrical water tank has height 8 m and radius 2m (see figure).

b. Is it true that it takes half as much work to pump the water out of the tank when it is half full as when it is full? Explain.

Textbook Question

Deceleration A car slows down with an acceleration of a(t) = −15 ft/s². Assume v(0)=60 ft/s,s(0)=0, and t is measured in seconds.


b. How far does the car travel in the time it takes to come to rest?

Textbook Question

A torus (doughnut) A torus is formed when a circle of radius 2 centered at (3, 0) is revolved about the y-axis.


b. Use the washer method to write an integral for the volume of the torus.

Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.


b. When the velocity is positive on an interval, the displacement and the distance traveled on that interval are equal.

Textbook Question

Functions from arc length What differentiable functions have an arc length on the interval [a, b] given by the following integrals? Note that the answers are not unique. Give a family of functions that satisfy the conditions.

b. ∫a^b √1+36 cos² 2xdx