Skip to main content
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 6.7.36b
Chapter 6, Problem 6.7.36b

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.
<IMAGE>

Verified step by step guidance
1
Understand that the work required to pump water out of the tank depends on both the volume of water and the distance each layer of water must be lifted.
Set up the integral for work when the tank is full: consider a thin horizontal slice of water at height \(y\) with thickness \(dy\). The volume of this slice is the area of the base times \(dy\), which is \(\pi \times 2^2 \times dy = 4\pi dy\).
Calculate the weight of this slice by multiplying its volume by the density of water and gravity (these constants can be factored out). The distance the slice must be lifted is \(8 - y\) meters (from height \(y\) to the top of the tank). So, the work done on this slice is proportional to \(4\pi (8 - y) dy\).
Integrate this expression from \(y=0\) to \(y=8\) to find the total work when the tank is full: \(W_{full} = \int_0^8 4\pi (8 - y) dy\).
For the tank half full, repeat the process but integrate from \(y=0\) to \(y=4\): \(W_{half} = \int_0^4 4\pi (8 - y) dy\). Compare \(W_{half}\) to \(\frac{1}{2} W_{full}\) to determine if it takes half as much work to pump the water out when the tank is half full.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
10m
Was this helpful?

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 calculated as the integral of force over distance. When pumping water from a tank, the force varies with the weight of the water at different heights, requiring integration to sum the work done on each layer of water.
Recommended video:
05:40
Work Done On A Spring (Hooke's Law)

Volume and Weight of Water in a Cylinder

The volume of water in a cylindrical tank depends on the height of the water. Since weight is proportional to volume, the amount of work depends on how much water is present, which changes as the tank empties.
Recommended video:
04:48
Finding Volume Using Disks

Distance Water is Pumped

The work depends not only on the amount of water but also on the distance each layer of water must be lifted. Water at the bottom must be pumped higher than water near the top, so the work is not simply proportional to the volume.
Recommended video:
07:39
Pumping Liquids
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?

1
views
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

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.

1
views
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

Textbook Question

Determine whether the following statements are true and give an explanation or counterexample. 


b. If f is not one-to-one on the interval [a, b], then the area of the surface generated when the graph of f on [a, b] is revolved about the x-axis is not defined.