Concept explainers
(a)
To explain: Thereason for the circumference of the base of the cone is
(a)
Answer to Problem 55E
The circumference of the base of the cone is
Explanation of Solution
Given information:A sector of arc length x is cut out from a circular piece of paper with radius 4 in to formed a cone with radius r and height h as shown below.
Figure (1)
Calculation:
The formula for the circumference of a circle with radius r is:
Substitute 4 for r in the above formula to find the circumference of given circle,
It is given that the circumference of the base of the cone is
A sector of arc length x is cut out from the circle to form the cone. So, the circumference of the base of the cone is the arc length of sector subtracted from the circumference of the circle.
Therefore, the circumference of the base of the cone is
(b)
To express: The radius of the cone as a function of x .
(b)
Answer to Problem 55E
The radius as a function of x can be written as
Explanation of Solution
Given information: A sector of arc length x is cut out from a circular piece of paper with radius 4 in to formed a cone with radius r and height h as shown below.
Figure (1)
Calculation:
It is given that the circumference of the base of the cone is
The formula for the circumference of a circle with radius r is:
Substitute
Therefore, the radius as a function of x can be written as
(c)
To express: The height h of the cone as a function of x .
(c)
Answer to Problem 55E
The height h of the cone as a function x can be written as
Explanation of Solution
Given information: A sector of arc length x is cut out from a circular piece of paper with radius 4 in to formed a cone with radius r and height h as shown below.
Figure (1)
Calculation:
From part (b), the radius as a function of x can be written as
The formula for the slant height l of a cone is:
Substitute 4 for land
Further simplify.
Therefore, the height h of the cone as a function x can be written as
(c)
To express: The volume V of the cone as a function of x .
(c)
Answer to Problem 55E
The volumeV of the cone as a function x can be written as
Explanation of Solution
Given information: A sector of arc length x is cut out from a circular piece of paper with radius 4 in to formed a cone with radius r and height h as shown below.
Figure (1)
Calculation:
From part (b), the radius of the cone is
The formula for the volumeV of a cone is:
Substitute
Therefore, the volume V of the cone as a function x can be written as
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Calculus 2012 Student Edition (by Finney/Demana/Waits/Kennedy)
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