top button
Flag Notify
    Connect to us
      Facebook Login
      Site Registration

Facebook Login
Site Registration

What is the maximum volume of a cone inscribed in a sphere of radius 6?

+1 vote
117 views

What is the maximum volume of a cone inscribed in a sphere of radius 6?

Note: share your workings also

posted May 10, 2018 by anonymous

Share this puzzle
Facebook Share Button Twitter Share Button LinkedIn Share Button

2 Answers

0 votes
answer May 10, 2018 by Tejas Naik
0 votes

Let H = height of the cone , R = radius of the base of the cone and X = distance from the centre of the sphere to to the base of the cone.
We can write the following equation:
R( exp2) + X (exp2) = 6 (exp2) (1)
The volume of the cone is given by the equation:
V = 1/3 PI R(exp2) H (2)
then, we have to maximize : R(exp2) H
From the equation (1), R(exp2) = 36 - X(exp2)
In addition, H = 6 + X
Therefore we obtain the function:
F(X) = (6 + X) ( 36 - X(exp2) or F(x) = 216 - 6 X(exp2) + 36 X - X(exp3)
The derivative of the function , d(F(X) /d X = -12 X + 36 - 3 X(exp2) This is a quadratic equation
At the maximum value for, a given X, the derivative equals cero
Then we have to solve the quadratic equation which yields two values X = 2 , X = -6
We take the value 2,
The cone of maximum volume has a height of 8 and a radius of sqrt ( 32)
By replacing values into the equation 2 , we obtain V = 268

answer May 14, 2018 by anonymous



Similar Puzzles
0 votes

A circular sector can be folded into a cone by joining its two radii.
For a circular sector with a radius equal to 1, what is the maximum volume of the cone? If the circular sector has a central angle θ, what is the value of θ at the maximum volume?

enter image description here

0 votes

In a square with a side length of 1, two quarter circles are drawn and a circle is inscribed between the quarter circles, as shown in the diagram.
What is the radius of the inscribed circle?

enter image description here

+2 votes

In a circle of radius 1, an equilateral triangle is inscribed in the circle as shown. What is the area of the blue region?

What is the area of the blue region

Contact Us
+91 9880187415
sales@queryhome.net
support@queryhome.net
#280, 3rd floor, 5th Main
6th Sector, HSR Layout
Bangalore-560102
Karnataka INDIA.
QUERY HOME
...