A cone is inscribed in a sphere of diameter 32 cm. Angle between height and slant height of the cone is 30. Find the volume of the cone.
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To find the volume of the cone, we need to know its height and radius. Since the cone is inscribed in a sphere, we can use the properties of the sphere to find these values.
Given that the diameter of the sphere is 32 cm, we know that the radius of the sphere is half of its diameter, which is 32/2 = 16 cm.
Since the cone is inscribed in the sphere, its height will be equal to the radius of the sphere. Therefore, the height of the cone is 16 cm.
Now, let's find the slant height of the cone. We are given that the angle between the height and the slant height of the cone is 30 degrees. To find the slant height, we can use the trigonometric relationship of the right triangle formed by the height, slant height, and base of the cone.
In this case, we can use the sine function:
sin(angle) = opposite/hypotenuse.
Since the opposite side is the height of the cone and the hypotenuse is the slant height, we have:
sin(30°) = height/slant height.
Solving for the slant height:
slant height = height/sin(30°) = 16 cm / sin(30°).
Using a calculator, we can find the value of sin(30°) ≈ 0.5.
Therefore, the slant height of the cone is:
slant height = 16 cm / 0.5 ≈ 32 cm.
Now that we have the height and slant height of the cone, we can calculate its volume using the formula:
Volume = (1/3) * pi * radius^2 * height.
Plugging in the values:
Volume = (1/3) * π * (16 cm)^2 * 16 cm.
Calculating this expression:
Volume ≈ 10752 cm^3.
Therefore, the volume of the cone is approximately 10752 cubic centimeters.