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hw13solutions - Math 2250 Written HW#13 Solutions 1 Find...

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Math 2250 Written HW #13 Solutions 1. Find the volume of the largest right circular cone that can be inscribed in a sphere of radius 3. An illustration is given on p. 261, problem #12. Remember that the volume of a circular cone with radius r and height h is 1 3 πr 2 h . ( Hint: which variable you choose to express the volume in terms of makes a big difference in how easy it is to differentiate ) Answer: First, notice that, by the Pythagorean Theorem, x 2 + y 2 = 3 2 , meaning that x 2 = 9 - y 2 . Also, since the volume of a cone with radius r and height h is 1 3 πr 2 h , we know that the volume of the cone is 1 3 πx 2 (3 + y ) = 1 3 π (9 - y 2 )(3 + y ) = 1 3 π 27 + 9 y - 3 y 2 - y 3 . Therefore, we want to maximize the function V ( y ) = 1 3 π 27 + 9 y - 3 y 2 - y 3 subject to the constraint 0 y 3. To find the critical points, we differentiate: V 0 ( y ) = 1 3 π 9 - 6 y - 3 y 2 = π 3 - 2 y - y 2 = π (3 + y )(1 - y ) . Therefore, V 0 ( y ) = 0 when π (3 + y )(1 - y ) = 0 , meaning that y = - 3 or y = 1. Only y = 1 is in the interval [0 , 3], so that’s the only critical point we need to concern ourselves with.
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