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A solid right circular cone is inscribed inside a hollow sphere of radius 10 cm. The height of the cone is two times its radius (i.e., radius of base of the cone). Find the ratio of volume of the cone to the volume of the sphere.
- a)32 : 81
- b)64 : 125
- c)16 : 125
- d)32: 125
Correct answer is option 'D'. Can you explain this answer?
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A solid right circular cone is inscribed inside a hollow sphere of rad...
Let the height of the cone be 2r, where r is the radius of base of the cone.
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A solid right circular cone is inscribed inside a hollow sphere of rad...
To find the ratio of the volume of the cone to the volume of the sphere, we need to calculate the volume of each shape and then compare them.
Let's start by calculating the volume of the cone and the sphere separately.
Volume of the Cone:
Given that the height of the cone is two times its radius, we can say that the height (h) is equal to 2r.
The volume of a cone is given by the formula: V = (1/3)πr^2h.
Substituting the given values, we have V_cone = (1/3)πr^2(2r) = (2/3)πr^3.
Volume of the Sphere:
The radius of the sphere is given as 10 cm.
The volume of a sphere is given by the formula: V = (4/3)πr^3.
Substituting the given value, we have V_sphere = (4/3)π(10^3) = (4/3)π(1000) = (4000/3)π.
Now, we can calculate the ratio of the volume of the cone to the volume of the sphere.
Ratio = V_cone / V_sphere = ((2/3)πr^3) / ((4000/3)π) = (2/3)(r^3) / (4000/3) = (2/3)(r^3) * (3/4000) = (2/4000)(r^3) = (1/2000)(r^3).
Since we know that the height of the cone is two times its radius, we can substitute h = 2r into the formula for the volume of the cone.
Ratio = (1/2000)(r^3) = (1/2000)(2r)^3 = (1/2000)(8r^3) = (8/2000)(r^3) = (2/500)(r^3).
Now, we need to find the ratio of the volume of the cone to the volume of the sphere. Since the radius of the sphere is 10 cm, we can substitute r = 10 into the formula.
Ratio = (2/500)(10^3) = (2/500)(1000) = 4/5 = 32/40 = 32:40 = 32:125.
Therefore, the ratio of the volume of the cone to the volume of the sphere is 32:125, which is option D.
Let's start by calculating the volume of the cone and the sphere separately.
Volume of the Cone:
Given that the height of the cone is two times its radius, we can say that the height (h) is equal to 2r.
The volume of a cone is given by the formula: V = (1/3)πr^2h.
Substituting the given values, we have V_cone = (1/3)πr^2(2r) = (2/3)πr^3.
Volume of the Sphere:
The radius of the sphere is given as 10 cm.
The volume of a sphere is given by the formula: V = (4/3)πr^3.
Substituting the given value, we have V_sphere = (4/3)π(10^3) = (4/3)π(1000) = (4000/3)π.
Now, we can calculate the ratio of the volume of the cone to the volume of the sphere.
Ratio = V_cone / V_sphere = ((2/3)πr^3) / ((4000/3)π) = (2/3)(r^3) / (4000/3) = (2/3)(r^3) * (3/4000) = (2/4000)(r^3) = (1/2000)(r^3).
Since we know that the height of the cone is two times its radius, we can substitute h = 2r into the formula for the volume of the cone.
Ratio = (1/2000)(r^3) = (1/2000)(2r)^3 = (1/2000)(8r^3) = (8/2000)(r^3) = (2/500)(r^3).
Now, we need to find the ratio of the volume of the cone to the volume of the sphere. Since the radius of the sphere is 10 cm, we can substitute r = 10 into the formula.
Ratio = (2/500)(10^3) = (2/500)(1000) = 4/5 = 32/40 = 32:40 = 32:125.
Therefore, the ratio of the volume of the cone to the volume of the sphere is 32:125, which is option D.
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A solid right circular cone is inscribed inside a hollow sphere of radius 10 cm. The height of the cone is two times its radius (i.e., radius of base of the cone). Find the ratio of volume of the cone to the volume of the sphere.a)32 : 81b)64 : 125c)16 : 125d)32: 125Correct answer is option 'D'. Can you explain this answer?
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