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2 years ago

A cone is 10cm high and has a base radius of 8cm.

Mathematics

1 answer:

slavikrds [6]2 years ago

3 0

The maximum volume of the cylinder in exact form is; V_max = 2560π/27

<h3>How to maximize the volume of a cylinder?</h3>

Let us define the variables as:

Radius of cone = r

Radius of cylinder = x

Height of Cone = h

Height of Cylinder = y

The general equation for volume of the cylinder is;

V = πx²y

Taking a strip of both figures and relating x, y, r & h as well as using ratios of similar triangles, we have:

Height of triangle above cylinder/Base of Triangle above cylinder = Height of full triangle/Base of full triangle

This gives;

(h - y)/2x = h/2r

Making y the subject gives us;

h – y = hx/r

y = h – hx/r

y = h(1 – x/r)

Plug in y into our Volume equation:

V(x) = πx²h(1 – x/r)

V(x) = πh(x² - x³/r)

To get maximum volume, find the derivative of the volume and solve for when the derivative equals zero:

V'(x) = πh(2x - 3x²/r)

V'(x) = 0

Thus;

(2x - 3x²/r) = 0

x( 2r – 3x) = 0

Thus;

x = 0 or x = 2r/3

Put x = 2r/3 in our volume equation to find V_max

V_max = πh((2r/3)² – (2r/3)³/r)

V_max = πh(4r²/9 - 8r²/27)

V_max = πh(12r²/27 - 8r²/27)

V_max = 4πhr²/27

Thus, at r = 8 and h = 10, we have;

V_max = 4π(10)8²/27

V_max = 2560π/27

Read more about Maximizing Volume of Cylinder at; brainly.com/question/10373132

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