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
Please see the attached file for the complete solution.
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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,
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 ...
Maximum and minimum values are found using derivatives. The solution is detailed and well presented.