The volume of a cone is 25/3 * pi*c * m ^ 3 What is the volume of a sphere if its radius is the same as the cone's and the height of the cone is equal to the sphere's diameter?
25/6 * pi*cm ^ 3;
25/2 * pi*cm ^ 3;
25pi*cm ^ 3;
50/3 * pi*cm ^ 3
The volume of a cone is given by the formula V = (1/3) * pi * r^2 * h, where r is the radius and h is the height.
In this case, the volume of the cone is (25/3) * pi * c * m^3.
Since the radius of the cone is the same as the radius of the sphere, we can substitute the radius of the cone with the radius of the sphere in the formula for volume of the sphere.
The height of the cone is equal to the diameter of the sphere, which is twice the radius of the sphere. Therefore, the height of the cone is 2r.
Substituting the values into the formula for volume of the cone, we have:
(25/3) * pi * c * m^3 = (1/3) * pi * r^2 * 2r
Simplifying:
(25/3) * c * m^3 = (2/3) * r^3 * pi
Dividing both sides by (2/3) * pi, we have:
(25/3) * c * m^3 / ((2/3) * pi) = r^3
Taking the cube root of both sides to solve for r, we have:
r = [(25/3) * c * m^3 / ((2/3) * pi)]^(1/3)
Now, we can substitute this value of r into the formula for the volume of the sphere, which is (4/3) * pi * r^3.
Substituting r, we have:
Volume of the sphere = (4/3) * pi * [(25/3) * c * m^3 / ((2/3) * pi)]^(1/3) * [(25/3) * c * m^3)^2 / (2/3) * pi)]^(1/3)
Simplifying further:
Volume of the sphere = (4/3) * pi * [(25/3)^3 * (c * m^3)^3 / ((2/3) * pi)^3]^(1/3)
Cancelling out the pi terms, we have:
Volume of the sphere = (4/3) * [(25/3)^3 * (c * m^3)^3 / ((2/3))^3]^(1/3)
Simplifying the exponents inside the brackets:
Volume of the sphere = (4/3) * [(25^3 /3^3) * (c^3 * m^9) / (2^3 / 3^3)]^(1/3)
Cancelling out the powers of 3 in the numerator and denominator:
Volume of the sphere = (4/3) * [(25^3 / 2^3) * (c^3 * m^9)]^(1/3)
Simplifying the numerator:
Volume of the sphere = (4/3) * [(15625 / 8) * (c^3 * m^9)]^(1/3)
Rearranging the terms:
Volume of the sphere = (4/3) * (15625 / 8)^(1/3) * (c^3 * m^9)^(1/3)
Simplifying further:
Volume of the sphere = (4/3) * (15625 / 8)^(1/3) * (c * m^3)^(1/3) * m^2
The terms (15625 / 8)^(1/3) and (c * m^3)^(1/3) can be simplified:
(15625 / 8)^(1/3) = (15625)^(1/3) / (8)^(1/3) = 25 / 2
(c * m^3)^(1/3) = c^(1/3) * (m^3)^(1/3) = c^(1/3) * m
Substituting these simplified terms, we have:
Volume of the sphere = (4/3) * (25 / 2) * (c * m) * m^2
Simplifying further:
Volume of the sphere = (50/3) * (c * m) * m^2
Converting c * m to cm and simplifying the units, we have:
Volume of the sphere = (50/3) * pi * cm^3
Therefore, the volume of the sphere is 50/3 * pi * cm^3.
Answer: 50/3 * pi * cm^3.