Volume of cone with base radius 1cm is22√3/21cc.Find vertical angle of cone
volume=1/3 PI r^3*H
vertical angle = arcTan(H/r)
H/r=tan Theta
volume= 1/3 PI r^4 (tanTheta)
tanTheta=3*volume/(PI*r^4)
solve that.
I believe it's
v = 1/3 πr^2 h
so, 1/3 πr^2 h = 22√3/21
whatever that means 22√(3/21)? 22/21 √3? 22 ∛21?
and, as bobpursley said, h = r tanθ
To find the vertical angle of a cone, we can use trigonometry. The vertical angle is the angle formed between the height of the cone and the cone's axis.
First, let's find the height of the cone. We know that the volume of the cone is given as 22√3/21 cc.
The formula for the volume of a cone is given by V = (1/3) * π * r^2 * h, where V is the volume, r is the base radius, and h is the height.
Plugging in the values into the formula, we have:
22√3/21 = (1/3) * π * (1)^2 * h
Simplifying the equation, we get:
22√3/21 = π * h/3
To isolate h, we multiply both sides by 3 and divide by π:
h = (22√3/21) * (3/π)
Now let's calculate the height:
h = (22√3/21) * (3/π) ≈ 0.6411 cm
Next, let's find the slant height of the cone. The slant height (l) and the base radius (r) form a right-angle triangle with the height as the hypotenuse.
Using the Pythagorean theorem, we have:
l^2 = r^2 + h^2
Plugging in the values, we get:
l^2 = (1)^2 + (0.6411)^2
l^2 = 1 + 0.4114
l^2 ≈ 1.4114
Take the square root of both sides to find the slant height:
l ≈ √1.4114
l ≈ 1.1872 cm
Finally, to find the vertical angle (θ), we can use the tangent function:
tan(θ) = r/h
Plugging in the values, we have:
tan(θ) = 1/(0.6411)
θ = tan^(-1)(1/(0.6411))
Using a scientific calculator or an online calculator, we find that:
θ ≈ 56.31 degrees
Therefore, the vertical angle of the cone is approximately 56.31 degrees.