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# Geometry - cones

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A cone has a volume of 12288*pi cubic inches and the vertex angle of the vertical cross section is 60 degrees. To the nearest tenth, what is the height of the cone?

I tried various calculations using the Pythagorean Theorem and the 30-60-90 triangle rule, but I just don't get what I'm supposed to do with the given information. Please help!

Thanks!

May 2, 2023

#1
+1347
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The volume of a cone is given by the formula:

V = (1/3)πr^2h

Where:

V is the volume of the cone

r is the radius of the base of the cone

h is the height of the cone

We know that the volume of the cone is 12288*pi cubic inches and that the vertex angle of the vertical cross section is 60 degrees. This means that the cross section of the cone is an equilateral triangle.

The radius of the base of the cone is equal to the height of the equilateral triangle divided by the sine of 60 degrees.

r = h / sin(60)

r = h / (√3/2)

r = 2h / √3We can now plug this value of r into the formula for the volume of the cone:V = (1/3)π(2h/√3)^2hV = (4/3)πh^3We know that the volume of the cone is 12288*pi cubic inches, so we can set this equal to the right-hand side of the equation above:

12288*pi = (4/3)πh^3

h^3 = 3072

h = 17.7

To the nearest tenth, the height of the cone is 17.8 inches.

May 2, 2023
#2
+214
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Almost. However, when I cuberoot 3072, I get 14.5

Is that what you meant to say?

May 2, 2023