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Find the radius r of the right circular cone of maximum volume which can be inscribed in a sphere of radius R.
Solution:
To illustrate the problem, it is better to draw the figure as follows
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| Photo by Math Principles in Everyday Life |
We know that the volume of a right circular cone is
Next, we need another equation in order to eliminate h at the above equation. Apply Pythagorean Theorem at the right triangle inside the right circular cone, we have
Substitute the value of h to the first equation, we have
Take the derivative on both sides of the equation with respect to r. Consider R as a constant because a right circular cone is inscribed in a sphere.
Equate dV/dr = 0 because we want to maximize the volume of a right circular cone
Divide both sides of the equation by ⅓ πr, we have
Divide both sides of the equation by 2R, we have
Square on both sides of the equation to remove the radical sign, we have
Therefore,
