Applied optimization problem. Max volume of a cylinder in a cone
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Applied optimization problem. Max volume of a cylinder in a cone

[From: ] [author: ] [Date: 11-10-29] [Hit: ]
and the axis = + y axis.y = - 5/5.where x is between O and 5.5, and h = f(x) = - 5/5.--> V = pi*x^2(- 5/5.......
A cylinder is inscribed in a right circular cone of height 5 and radius (at the base) equal to 5.5. What are the dimensions of such a cylinder which has maximum volume?

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Hello

place the cone cross section so into the coordinate system, that the center of the bottom is at the origin, and the axis = + y axis.
the right slant line is
y = - 5/5.5*x + 5

The volume of the cylinder is V = pi*x^2*h
where x is between O and 5.5, and h = f(x) = - 5/5.5*x + 5
--> V = pi*x^2(- 5/5.5*x + 5)
to find the maximum set the first derivative = 0:
V' = 10/11*pi*(11x - 3x^2) = 0
--------------
the solution for x is
x = 3.6667 (= radius)
h = - 5/5.5*3.6667 + 5) = 1.666 (height)

and the maximum volume is
Vmax = pi*3.6667^2(-5/5.5*3.6667 +5)
Vmax = 70.3949 unit^3

Regards
1
keywords: of,Max,in,volume,problem,cylinder,cone,optimization,Applied,Applied optimization problem. Max volume of a cylinder in a cone
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