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?
-
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
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