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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 6.7.39a
Chapter 6, Problem 6.7.39a

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).
a. If the tank is full, how much work is required to pump the water to the level of the top of the tank and out of the tank?
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Identify the variables and set up the coordinate system: Let the vertical axis y measure the height from the tip (vertex) of the cone upwards, with y = 0 at the tip and y = 6 m at the top of the tank.
Express the radius of the water surface at height y: Since the tank is a cone, the radius r at height y varies linearly from 0 at y = 0 to 1.5 m at y = 6 m. Use similar triangles to write the radius as a function of y: \(r(y) = \frac{1.5}{6} y = 0.25 y\).
Write the volume of a thin horizontal slice of water at height y: The slice has thickness \(dy\) and radius \(r(y)\), so its volume is \(dV = \pi [r(y)]^2 dy = \pi (0.25 y)^2 dy = \pi \times 0.0625 y^2 dy\).
Determine the work required to pump this slice of water to the top: The weight density of water is \(\rho g\) (where \(\rho\) is the density of water and \(g\) is acceleration due to gravity). The distance the slice must be lifted is \(6 - y\). So the work to move this slice is \(dW = \rho g \times dV \times (6 - y)\).
Set up the integral for total work: Integrate \(dW\) from \(y = 0\) to \(y = 6\) to find the total work required to pump all the water out of the tank: \(W = \int_0^6 \rho g \pi (0.0625 y^2) (6 - y) dy\). This integral can then be evaluated 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 from a tank, the force varies with the weight of the water slice being moved and the distance it must be lifted. Calculating work involves integrating the product of force and displacement over the height of the tank.
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Volume of a Thin Slice of Water in a Cone

To find the work done, the tank is divided into thin horizontal slices of water. Each slice is a thin disk whose radius depends on its height in the cone. Using similar triangles, the radius at any height can be expressed as a function of height, allowing calculation of the slice's volume.
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Relationship Between Radius and Height in a Cone

The radius of the water slice changes linearly with height due to the conical shape. Using the given dimensions, the radius at height y can be found by proportionality: radius = (1.5/6) * y. This relationship is essential to express the volume and weight of each slice as functions of height.
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Related Practice
Textbook Question

Flow rates in the Spokane River The daily discharge of the Spokane River as it flows through Spokane, Washington, in April and June is modeled by the functions

r1(t) = 0.25t²+37.46t+722.47 (April) and

r2(t) = 0.90t²−69.06t+2053.12 (June), where the discharge is measured in millions of cubic feet per day, and t=0 corresponds to the beginning of the first day of the month (see figure).

a. Determine the total amount of water that flows through Spokane in April (30 days). 

Textbook Question

17–22. Position from velocity Consider an object moving along a line with the given velocity v and initial position.


a. Determine the position function, for t≥0, using the antiderivative method


v(t) = −t³+3t²−2t on [0, 3]; s(0)=4

Textbook Question

Critical depth A large tank has a plastic window on one wall that is designed to withstand a force of 90,000 N. The square window is 2 m on a side, and its lower edge is 1 m from the bottom of the tank.

a. If the tank is filled to a depth of 4 m, will the window withstand the resulting force?

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Textbook Question

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


a. A pyramid is a solid of revolution. "

Textbook Question

Oscillating growth rates Some species have growth rates that oscillate with an (approximately) constant period P. Consider the growth rate function N'(t) = r+A sin 2πt/P, where A and r are constants with units of individuals/yr, and t is measured in years. A species becomes extinct if its population ever reaches 0 after t=0.


a. Suppose P=10, A=20, and r=0. If the initial population is N(0)=10, does the population ever become extinct? Explain.

Textbook Question

For the given regions R₁ and R₂, complete the following steps.


a. Find the area of region R₁.


R₁ is the region in the first quadrant bounded by the y-axis and the curves y=2x^2 and y=3−x; R₂ is the region in the first quadrant bounded by the x-axis and the curves y=2x^2 and y=3−x(see figure).