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Derivatives : Maximum and Minimum Values

9). Find the dimensions of a right circular cone of minimum volume that can be circumscribed about a sphere of radius 8cm.

32 , 8root

11. The sum of 2 positive numbers is 20. Find the numbers if
A) their product is a maximum
B) the sum of their squares is a minimum
C) the product of the square root of one and the cube root of the other is a maximum

12). The product of 2 positive numbers is 16. Find the numbers if
A) their sum is a minimum
B) the sum of one and the square of the other is a minimum

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9). Find the dimensions of a right circular cone of minimum volume that can be circumscribed about a sphere of radius 8cm

32 , 8root

In the above figure, Let r be the radius of the base of the cone and let y + b be the height of the cone.
From the similar triangles ABC and AED,

Then
The volume of the cone is

Differentiate with respect to y

The only critical number is y = 3b so that dV/dy = 0.
when y < 3b, dV/dy <0, and y >3b, dV/dy >0, Using the first derivative test, at the above critical ...

Solution Summary

Maximum and minimum values are found using derivatives. The solution is detailed and well presented.

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