Mathematics, 05.10.2019 21:00, jayy1733
Find the volume of the largest right circular cone that can be inscribed in a sphere of radius 3?
Answers: 2
we know that the volume of the cone is:![\frac{1}{3} \pi x^2 (3+y) = \frac{1}{3} \pi (9-y^2)(3+y) = \frac{1}{3} \pi [27+9y-3y^2-y^3]](/tpl/images/0290/8604/a236e.png)
![V(y) = \frac{1}{3} \pi [27+9y-3y^2-y^3]](/tpl/images/0290/8604/97c25.png)
subject to the constraint 
.![V'(y)= \frac{1}{3} \pi [9-6y-3y^2] = \pi [3-2y-y^2] = \pi (3+y)(1-y).](/tpl/images/0290/8604/424c3.png)
when 
or 
. Only ![[0,3]](/tpl/images/0290/8604/a909d.png)
so that’s the only critical point we need to concern ourselves with. 
at the critical point and the endpoints:![V(0) = \frac{1}{3} \pi [27+9(0) - 3(0)^2] = 9 \pi](/tpl/images/0290/8604/f4ce2.png)
![V(1) = \frac{1}{3} \pi [27+9(1)-3(1)^2-1^3] = \frac{32 \pi }{3}](/tpl/images/0290/8604/9efc5.png)
![V(3) = \frac{1}{3} \pi [27+9(3) - 3(3)^2-3^2] = 0](/tpl/images/0290/8604/e6832.png)
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Find the volume of the largest right circular cone that can be inscribed in a sphere of radius 3?...
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