Find the largest possible volume of a cone that fits inside a sphere of radius one.

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Find the largest possible volume of a cone that fits inside a sphere of radius one.

Mathematics
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you get the volume of the sphere which is 4.18879
then you just have to find a volume of a cone smaller than that

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https://sketch.io/render/sketch55e116f6c4926.png
Use the above diagram and find an equation to represent the volume of the cone in terms of r. Then you can get the answer :-)
Find the volume of the largest right circular cone that can be inscribed in a sphere of radius 3? Let R = radius sphere r = base radius cone R + h = height cone V = volume cone _______ V = (1/3)πr²(R + h) By the Pythagorean Theorem: r² = R² - h² Plug into the formula for volume. V = (1/3)π(R² - h²)(R + h) = (1/3)π(R³ + R²h - Rh² - h³) Take the derivative and set equal to zero to find the critical points. dV/dh = (1/3)π(R² - 2Rh - 3h²) = 0 R² - 2Rh - 3h² = 0 (R - 3h)(R + h) = 0 h = R/3, -R But h must be positive so: h = R/3 Calculate the second derivative to determine the nature of the critical points. d²V/dh² = (π/3)(-2R - 6h) < 0 So this is a relative maximum which we wanted. Solve for r². r² = R² - h² = R² - (R/3)² = R²(1 - 1/9) = (8/9)R² Calculate maximum volume. V = (π/3)[(8/9)R²](R + R/3) = (8/27)πR³(4/3) = 32πR³/81 For R = 3 maximum volume is: V = 32π(3³)/81 = 32π(27)/81 = 32π/3
this was the question that came in our examination and this is how i solved it ....and this is similar to your problem only difference is that you gotta put 1 in place 3 (radius):))))))

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