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A cone of maximum volume is inscribed in a given sphere. Then the ratio of the height of the cone to the diameter of the sphere is
- a)2/3
- b)3/4
- c)1/3
- d)1/4
Correct answer is option 'A'. Can you explain this answer?
Most Upvoted Answer
A cone of maximum volume is inscribed in a given sphere. Then the rati...
Insight into the problem:
The maximum volume of a cone inscribed in a sphere occurs when the cone's base is the same as the sphere's diameter. Let's denote the height of the cone as h and the diameter of the sphere as d.
Solution:
- Let the radius of the sphere be 'r' and the radius of the base of the cone be 'x'.
- The height of the cone, using similar triangles, can be expressed as h = 2r - x.
- The volume of the cone V = (1/3) * π * x^2 * h.
- Substituting h = 2r - x into the volume formula, V = (1/3) * π * x^2 * (2r - x).
- To maximize V, we differentiate V w.r.t. x and set it to zero.
- dV/dx = 2πx(2r - 2x) = 0.
- Solving for x, x = r.
- The height of the cone, h = 2r - r = r.
- Therefore, the ratio of the height of the cone to the diameter of the sphere is h/d = r/(2r) = 1/2 = 2/3.
Therefore, the correct answer is option 'A' (2/3).
The maximum volume of a cone inscribed in a sphere occurs when the cone's base is the same as the sphere's diameter. Let's denote the height of the cone as h and the diameter of the sphere as d.
Solution:
- Let the radius of the sphere be 'r' and the radius of the base of the cone be 'x'.
- The height of the cone, using similar triangles, can be expressed as h = 2r - x.
- The volume of the cone V = (1/3) * π * x^2 * h.
- Substituting h = 2r - x into the volume formula, V = (1/3) * π * x^2 * (2r - x).
- To maximize V, we differentiate V w.r.t. x and set it to zero.
- dV/dx = 2πx(2r - 2x) = 0.
- Solving for x, x = r.
- The height of the cone, h = 2r - r = r.
- Therefore, the ratio of the height of the cone to the diameter of the sphere is h/d = r/(2r) = 1/2 = 2/3.
Therefore, the correct answer is option 'A' (2/3).
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A cone of maximum volume is inscribed in a given sphere. Then the ratio of the height of the cone to the diameter of the sphere isa)2/3b)3/4c)1/3d)1/4Correct answer is option 'A'. Can you explain this answer?
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