# volume problem.Find the maximum volume of a right circular cylinder inscribed in a cone of altitude 12 cm and base radius 4 cm, if the axis of two coincide.

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The volume of the cyllinder is:

V = A base*height

We'll write the volume of the cylinder as a function of x.

The area of the base of the cyllinder is:

A = pi*r^2

We'll determine r using proportions:

R/H = r/(12-x)

R = 4, H = 12, h is the height of the cylinder and r is the radius of the cylinder.

4/12 = r/(12-x)

1/3 = r/(12-x)

We'll cross multiply and we'll get:

3r = 12 - x

r = (12-x)/3

The volume of the cylinder is:

V(x) = pi*x*(12-x)^2/9

V(x) = (pi/9)*(144x - 24x^2 + x^3)

The volume is maximum for the critical value of x that cancels the first derivative of V(x).

We'll determine V'(x):

V'(x) = (pi/9)*(144 - 48x + 3x^2)

V'(x) = 0

144 - 48x + 3x^2 = 0

We'll divide by 3:

x^2 - 16x + 48 = 0

We'll apply quadratic formula:

x1 = [16+sqrt(256 - 192)]/2

x1 = (16+8)/2

x1 = 12 cm

x2 = (16-8)/2

x2 = 4 cm

For x = 12 cm, the volume of the cylinder is maximum.