the volume of a cone is 25/3 pi^3 what is the volume of a sphere if its radius is the same as the cones in the height of the cone is equal to the sphere's diameter ?
25/2 pi^3
50/3 pi^3
25/6 pi^3
25 pi^3
To find the volume of a sphere, we need to use the formula V = (4/3)πr³, where r is the radius of the sphere. In this case, the radius of the sphere is the same as the radius of the cone, and the height of the cone is equal to the diameter of the sphere.
Given that the volume of the cone is 25/3 π^3, we can find the radius using the formula V = (1/3)πr²h, where V is the volume, r is the radius, and h is the height.
Using the volume of the cone, we have (1/3)πr²h = 25/3 π^3.
Simplifying, we have r²h = 25 π^2.
Since the height of the cone is equal to the diameter of the sphere, we can write h = 2r.
Substituting into the equation, we have r²(2r) = 25 π^2.
This simplifies to 2r³ = 25 π^2.
Dividing both sides by 2, we get r³ = 25 π^2 / 2.
Finally, taking the cube root of both sides to solve for r, we have r = (25 π^2 / 2)^(1/3).
Now that we know the radius of the sphere, we can find its volume using the formula V = (4/3)πr³.
Substituting the value of r into the equation, we have V = (4/3)π[(25 π^2 / 2)^(1/3)]³.
Simplifying this expression gives V = (4/3)π(25 π^2 / 2)^(3/3).
Simplifying further, we have V = (25/6)π^3.
Therefore, the volume of the sphere is 25/6 π^3.
A sculptor created a design by carving a cone out of a cylinder The cone and cylinder share the same radius and height If the volume of cylinder before removing the cone is 54in .^ 3 what is the volume of the amount remaining?
27 in^3
40in .^ 3
36in .^ 3
18 ln^ 3
The volume of a cylinder is given by V = πr^2h, where r is the radius and h is the height. In this case, the volume of the cylinder before removing the cone is 54 in^3.
To find the volume of the cone, we need to subtract it from the volume of the cylinder. Let's call the radius and height of the cylinder r and h respectively.
The volume of the cylinder is 54 in^3, which means V_cylinder = πr^2h = 54.
Now, let's find the volume of the cone. The volume of a cone is given by V = (1/3)πr^2h, and since the cone and cylinder share the same radius and height, the cone's volume is (1/3)πr^2h.
The remaining volume after removing the cone is then V_remaining = V_cylinder - V_cone = 54 - (1/3)πr^2h.
Substituting the values, we have V_remaining = 54 - (1/3)πr^2h.
Since the cone and cylinder share the same radius and height, we can simplify the expression further. Let's substitute the expression for V_cylinder = 54 in^3.
V_remaining = 54 - (1/3)πr^2h = 54 - (1/3)(54) = 54 - 18 = 36 in^3.
Therefore, the volume of the amount remaining after removing the cone is 36 in^3.