Find the height of a cone with a volume of 138 cubic meters and a base area of 46 square meters
V = A * h
h = V / A = 138 / 46 = 3 m
Volume of a cone:
V = ( 1 / 3 ) * r ^ 2 * pi * h
r ^ 2 * pi = B
The area of the circle base
V = ( 1 / 3 ) * B * h
V = 138 m ^ 3
B = 46 m ^ 2
138 = ( 1 / 3 ) * 46 * h Multiply both sides by 3
3 * 138 = 1 * 46 * h
414 = 46 h Divide both sides by 46
414 / 46 = h
h = 414 / 46
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Remark:
414 = 2 * 3 * 3 * 23
46 = 2 * 23
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h = 2 * 3 * 3 * 23 / ( 2 * 23 )
h = 3 * 3
h = 9 m
To find the height of a cone, you can use the formula for the volume of a cone, which is given by:
V = 1/3 * π * r^2 * h,
where V is the volume, r is the radius of the base, and h is the height.
In this case, you are given the volume V and the base area A. The base area is related to the radius by the formula:
A = π * r^2.
To find the radius, rearrange the formula as:
r = √(A / π).
Now, substitute this value for r in the volume formula:
V = 1/3 * π * (√(A / π))^2 * h.
Simplify the formula by canceling π:
V = 1/3 * (√(A / π))^2 * h.
Now, solve the formula for h. Rearrange the formula as:
V = 1/3 * A * h.
Multiply both sides by 3:
3V = A * h.
Finally, solve for h:
h = 3V / A.
Substituting the given values, the height of the cone is:
h = 3 * 138 / 46 = 9 meters.
To find the height of a cone, you can use the formula for the volume of a cone and the formula for the base area of a cone. Here's how you can calculate it:
First, let's use the formula for the volume of a cone:
Volume of a cone = (1/3) * π * r^2 * h
where r is the radius of the base and h is the height of the cone.
You mentioned that the volume of the cone is 138 cubic meters. So, we can rewrite the formula as:
138 = (1/3) * π * r^2 * h
Next, let's use the formula for the base area of a cone:
Base area of a cone = π * r^2
You mentioned that the base area of the cone is 46 square meters. So, we can rewrite the formula as:
46 = π * r^2
Now, we have a system of two equations with two variables (r and h):
Equation 1: 138 = (1/3) * π * r^2 * h
Equation 2: 46 = π * r^2
We can rearrange Equation 2 to solve for r:
r^2 = 46 / π
r = sqrt(46 / π)
Substituting this value of r into Equation 1, we can solve for h:
138 = (1/3) * π * (sqrt(46 / π))^2 * h
Simplifying further, we get:
138 = (1/3) * (46 / π) * h
Multiply both sides of the equation by (3 * π) to isolate h:
138 * (3 * π) = 46 * h
h = (138 * 3 * π) / 46
Calculating this expression will give you the value of h, which is the height of the cone.