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
To find the volume of the cone, we need to know the height and the radius of the cone.
First, let's visualize the cone inscribed in the sphere:
The diameter of the sphere is given as 32 cm, which means the radius of the sphere is half of that, i.e., 32/2 = 16 cm.
Since the cone is inscribed in the sphere, the height of the cone is equal to the diameter of the sphere. Therefore, the height of the cone is also 32 cm.
Now, we need to find the radius of the cone.
Let's draw a diagram to understand the relationship between the height, slant height, and radius of the cone:
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/ | <- Slant Height (l)
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/ |
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/________________|
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Height (h)
From the diagram, we can see that the slant height (l), height (h), and radius (r) form a right-angle triangle.
The angle between the height and the slant height of the cone is given as 30 degrees. Since the trigonometric ratios are based on right-angle triangles, we need to work with the angles in the triangle to find the radius.
Since the angle between the height and slant height is 30 degrees, we can use the sine function to relate the height and the slant height:
sin(30) = h / l
To determine the value of sin(30), we can use a trigonometric table or a calculator.
sin(30) = 0.5
0.5 = h / l
Since we know that the height (h) is 32 cm, we can substitute this value into the equation to solve for the slant height (l):
0.5 = 32 / l
Rearranging the equation, we get:
l = 32 / 0.5
l = 64 cm
Now, we can calculate the radius (r) of the cone using the Pythagorean theorem:
r^2 = l^2 - h^2
r^2 = 64^2 - 32^2
r^2 = 4096 - 1024
r^2 = 3072
Taking the square root of both sides, we get:
r = sqrt(3072)
r ≈ 55.42 cm
Now that we have the height and the radius of the cone, we can calculate its volume using the formula:
Volume of a cone = (1/3) * π * r^2 * h
Volume = (1/3) * 3.14 * (55.42)^2 * 32
Volume ≈ 102,829.44 cubic cm
Therefore, the volume of the cone is approximately 102,829.44 cubic cm.