Conical Cup A conical cup is made from a circular piece of paper with radius 6 cm by cutting out a sector and joining the edges as shown below. Suppose θ = 5π/3.
- (a) Find the circumference C of the opening of the cup.
- (b) Find the radius r of the opening of the cup. [Hint: Use C = 2πr.]
- (c) Find the height h of the cup. [Hint: Use the Pythagorean Theorem.]
- (d) Find the volume of the cup.
(a)
The circumference
Answer to Problem 87E
The circumference
Explanation of Solution
Given:
Radius of the circular piece of paper is
Angle of the remaining circle is
Formulas used:
Circumference of circle is
Calculations:
The angle of a circle is
Substitute
The angle of a sector is
Formula for arc length is
Substitute
The arc length of the sector that has been cut out of the circle is
Circumference of the circle is
The complete circle has circumference as
Circumference of the opening of the conical cup is
Substitute
Thus, the circumference
(b)
The radius
Answer to Problem 87E
The radius
Explanation of Solution
From part (a),
Substitute
Thus, the radius
(c)
The height
Answer to Problem 87E
The height
Explanation of Solution
Given:
Radius of the circular piece of paper is
Angle of the remaining circle is
Slant height of the cone is
Calculation:
The Pythagorean Theorem is
The formula for the Pythagorean Theorem is
From part (b),
Substitute
Thus, the height
(d)
The volume of the cup.
Answer to Problem 87E
The volume of the cup is
Explanation of Solution
Formula used:
Volume of a cone is
Calculation:
From part (b) and (c),
Substitute
Thus, the volume of the cup is
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