Concept explainers
Conical Cup In this exercise we find the volume of the conical cup in Exercise 93 for any angle θ.
- (a) Follow the steps in Exercise 93 to show that the volume of the cup as a function of θ is
- (b) Graph the function V.
- (c) For what angle θ is the volume of the cup a maximum?
(a)
To show: The volume of the cup as a function of
Explanation of Solution
From Exercise
The radius and height of the cone to be taken in terms of
Formula for arc length is
Substitute
The arc length is
Circumference of the circle is
The radius of the circle in terms of
Formula for the Pythagorean Theorem is
Substitute
The height of the cone in terms of
Formula for the volume of cone is
Substitute
Hence, the volume of the cup as a function of
(b)
To sketch: A graph of the function
Explanation of Solution
The given function for volume of the cup is
The domain for
Graph
From Figure 1, the graph is in shape of sharp parabolas between
(c)
The angle
Answer to Problem 88E
The volume of the cup is maximum at
Explanation of Solution
From Figure 1, note that the cup has maximum volume 87.062 at 5.13. That is, when
Thus, the volume of the cup is maximum at
Chapter 6 Solutions
Precalculus: Mathematics for Calculus - 6th Edition
- Calculus: Early TranscendentalsCalculusISBN:9781285741550Author:James StewartPublisher:Cengage LearningThomas' Calculus (14th Edition)CalculusISBN:9780134438986Author:Joel R. Hass, Christopher E. Heil, Maurice D. WeirPublisher:PEARSONCalculus: Early Transcendentals (3rd Edition)CalculusISBN:9780134763644Author:William L. Briggs, Lyle Cochran, Bernard Gillett, Eric SchulzPublisher:PEARSON
- Calculus: Early TranscendentalsCalculusISBN:9781319050740Author:Jon Rogawski, Colin Adams, Robert FranzosaPublisher:W. H. FreemanCalculus: Early Transcendental FunctionsCalculusISBN:9781337552516Author:Ron Larson, Bruce H. EdwardsPublisher:Cengage Learning