To solve this problem, we need to find the formula for the volume of a cone. The volume of a cone is given by the formula V = (1/3)πr²h, where V is the volume, r is the radius, and h is the height.
In this case, the radius of the cone is the same as the sphere's radius. Let's call this common radius r. The height of the cone is equal to the sphere's diameter, which is twice the radius. So we can say that h = 2r.
Substituting these values into the cone volume formula, we get:
V = (1/3)πr²(2r)
V = (2/3)πr³
Since we are given the volume of the sphere as 72 m³, we can set up the equation:
72 = (2/3)πr³
To find the volume of the cone, we need to solve for r³. Rearranging the equation, we get:
r³ = (3/2) * (72/π)
r³ = 36/π
r = (36/π)^(1/3)
Now we can substitute this value of r into the cone volume formula to find the volume of the cone:
V = (2/3)π[(36/π)^(1/3)]³
V = (2/3)π * (36/π)
V = (2/3) * 36
V = 24 m³
Therefore, the volume of the cone is 24 m³.
The volume of a sphere is 72 m3 . What is the volume of a cone if its radius is the same as the sphere’s and its height is equal to the sphere’s diameter?(1 point)
m3
3 answers
wrong
Apologies for the mistake. Let's correct the solution.
The formula for the volume of a sphere is V = (4/3)πr³, where V is the volume and r is the radius.
Given that the volume of the sphere is 72 m³, we can set up the equation:
72 = (4/3)πr³
Simplifying the equation, we have:
r³ = (3/4) * (72/π)
r³ = 54/π
r = (54/π)^(1/3)
Now we need to find the volume of the cone. The formula for the volume of a cone is V = (1/3)πr²h, where h is the height.
Given that the height of the cone is equal to the sphere's diameter, which is twice the radius, we have:
h = 2r
The volume of the cone can be calculated as follows:
V = (1/3)πr²h
V = (1/3)π(54/π)^(1/3)²(2(54/π)^(1/3))
V = (1/3)π(54/π)^(2/3)(2(54/π)^(1/3))
Simplifying further, we have:
V = (1/3)π(2916/π²)(108/π)^(1/3)
V = (1/3)(2916/π)(108/π)^(1/3)
V = (972/π)(108/π)^(1/3)
V = 972(108/π²)^(1/3)
Using a calculator to evaluate (108/π²)^(1/3), we get approximately 4.857.
Therefore, the volume of the cone is approximately:
V = 972 * 4.857
V ≈ 4724.724 m³
The formula for the volume of a sphere is V = (4/3)πr³, where V is the volume and r is the radius.
Given that the volume of the sphere is 72 m³, we can set up the equation:
72 = (4/3)πr³
Simplifying the equation, we have:
r³ = (3/4) * (72/π)
r³ = 54/π
r = (54/π)^(1/3)
Now we need to find the volume of the cone. The formula for the volume of a cone is V = (1/3)πr²h, where h is the height.
Given that the height of the cone is equal to the sphere's diameter, which is twice the radius, we have:
h = 2r
The volume of the cone can be calculated as follows:
V = (1/3)πr²h
V = (1/3)π(54/π)^(1/3)²(2(54/π)^(1/3))
V = (1/3)π(54/π)^(2/3)(2(54/π)^(1/3))
Simplifying further, we have:
V = (1/3)π(2916/π²)(108/π)^(1/3)
V = (1/3)(2916/π)(108/π)^(1/3)
V = (972/π)(108/π)^(1/3)
V = 972(108/π²)^(1/3)
Using a calculator to evaluate (108/π²)^(1/3), we get approximately 4.857.
Therefore, the volume of the cone is approximately:
V = 972 * 4.857
V ≈ 4724.724 m³
1 answer