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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 6.3.69a
Chapter 6, Problem 6.3.69a

A right circular cylinder with height R and radius R has a volume of VC=πR^3 (height = radius).


a. Find the volume of the cone that is inscribed in the cylinder with the same base as the cylinder and height R. Express the volume in terms of VC.

Verified step by step guidance
1
Identify the given dimensions: The cylinder has height \(R\) and radius \(R\), so its volume is given by \(V_C = \pi R^3\).
Recall the formula for the volume of a cone: \(V_{cone} = \frac{1}{3} \pi r^2 h\), where \(r\) is the radius of the base and \(h\) is the height.
Since the cone is inscribed in the cylinder with the same base and height, set \(r = R\) and \(h = R\) in the cone volume formula.
Substitute these values into the cone volume formula to get \(V_{cone} = \frac{1}{3} \pi R^2 R = \frac{1}{3} \pi R^3\).
Express the cone volume in terms of the cylinder volume \(V_C\): since \(V_C = \pi R^3\), then \(V_{cone} = \frac{1}{3} V_C\).

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Key Concepts

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

Volume of a Cylinder

The volume of a right circular cylinder is calculated by multiplying the area of its circular base by its height. Specifically, the formula is V = πr²h, where r is the radius and h is the height. In this problem, both radius and height are equal to R, so the volume simplifies to V = πR³.
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Finding Volume Using Disks

Volume of a Cone

The volume of a cone is one-third the volume of a cylinder with the same base and height. The formula is V = (1/3)πr²h, where r is the radius of the base and h is the height. This relationship is essential for finding the cone's volume inscribed in the cylinder.
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Finding Volume Using Disks

Expressing Volume in Terms of Another Variable

Expressing one volume in terms of another involves substituting known values and simplifying. Here, the cone's volume should be expressed in terms of VC, the cylinder's volume, by using the relationship between their volumes and the given dimensions.
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Finding Volume Using Disks Example 3
Related Practice
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.


a. Determine when the motion is in the positive direction and when it is in the negative direction. 


v(t) = 50e^−2t on [0, 4]

Textbook Question

21–30. {Use of Tech} Arc length by calculator


a. Write and simplify the integral that gives the arc length of the following curves on the given interval. 

y = 1/x, for 1 ≤ x ≤ 10

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)=200−0.05x

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) = 6−2t on [0, 5]; s(0)=0

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Textbook Question

Region R is revolved about the line x=4 to form a solid of revolution.


a. What is the radius of a cross section of the solid at a point y in [1, 3]?

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.


a. Use the shell method to write an integral for the volume of the torus.