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? (1 point)
25/6 * pi*c * m ^ 3;
25/2 * pi*c * m ^ 3;
25pi*c * m ^ 3;
50/3 * pi*c * m ^ 3
The volume of a sphere is given by the formula V = (4/3) * pi * r^3, where r is the radius of the sphere.
Since the radius of the sphere is the same as the cone's, we can say that r = m.
The height of the cone is equal to the sphere's diameter, which means h = 2r = 2m.
Now, let's substitute these values into the formula for the volume of the cone:
V_cone = (25/3) * pi * c * m^3
Since the height of the cone is equal to 2r, we can rewrite the formula as:
V_cone = (25/3) * pi * c * (h/2)^3
Simplifying further:
V_cone = (25/3) * pi * c * (1/8) * h^3
Now, we can substitute this into the formula for the volume of the sphere:
V_sphere = (4/3) * pi * r^3
V_sphere = (4/3) * pi * (m^3)
Since the cone's volume is equal to the sphere's volume, we can set these two equations equal to each other:
(25/3) * pi * c * (1/8) * h^3 = (4/3) * pi * (m^3)
Simplifying further:
(25/3) * (1/8) * h^3 = (4/3) * (m^3)
(25/24) * h^3 = (4/3) * (m^3)
Multiplying both sides by (24/25):
h^3 = (4/3) * (m^3) * (24/25)
h^3 = (96/75) * (m^3)
h^3 = (32/25) * (m^3)
Taking the cube root of both sides:
h = (32/25)^(1/3) * (m)
So, the formula for the volume of the sphere with the given conditions is:
V_sphere = (4/3) * pi * (m^3) = (4/3) * pi * [(32/25)^(1/3) * (m)]^3 = (4/3) * pi * (32/25) * (m^3)
Simplifying:
V_sphere = (32/25) * (4/3) * pi * (m^3) = (32/25) * (4/3) * pi * (m^3) = (128/75) * pi * (m^3)
Thus, the correct answer is 128/75 * pi * c * m^3.
That’s not a answer
Apologies for the confusion. Let's go through the problem again.
The volume of a cone is given by the formula V_cone = (1/3) * pi * r^2 * h, where r is the radius of the cone and h is the height of the cone.
Given that the volume of the cone is (25/3) * pi * c * m^3, we can set up the equation as:
(25/3) * pi * c * m^3 = (1/3) * pi * r^2 * h
Since the radius of the cone is equal to the radius of the sphere, we can say that r = m.
And since the height of the cone is equal to the diameter of the sphere, we can say that h = 2r = 2m.
Substituting these values into the equation, we get:
(25/3) * pi * c * m^3 = (1/3) * pi * m^2 * (2m)
Simplifying:
(25/3) * pi * c * m^3 = (1/3) * pi * 2m^3
Cross-multiplying, we get:
(25/3) * c * m^3 = (2/3) * m^3
Dividing both sides by (2/3) * m^3, we get:
(25/3) * c = 1
Simplifying further:
25c = 3
Dividing both sides by 25, we get:
c = 3/25
Now, we can substitute this value back into the volume of the cone equation to find the volume of the sphere:
V_sphere = (4/3) * pi * r^3
V_sphere = (4/3) * pi * (m^3)
V_sphere = (4/3) * pi * [(2/3) * m^3]
V_sphere = (8/9) * pi * m^3
So the answer is (8/9) * pi * m^3.
To find the volume of the sphere, you can use the formula for the volume of a cone and apply it to the sphere.
Given that the volume of the cone is 25/3 * pi * c * m^3, we first need to find the radius (r) and height (h) of the cone.
The formula for the volume of a cone is V_cone = 1/3 * pi * r^2 * h.
In this case, we know that the height of the cone (h) is equal to the diameter of the sphere since they are equivalent.
So, h = 2r.
Now let's rearrange the formula for the volume of a cone to solve for the radius:
V_cone = 1/3 * pi * r^2 * h
25/3 * pi * c * m^3 = 1/3 * pi * r^2 * 2r
25 * c * m^3 = r^2 * 2r
Now we can solve for r:
25 * c * m^3 = 2r^3
r^3 = (25 * c * m^3) / 2
r = (25 * c * m^3)^(1/3) / ∛2
Now that we have the radius of the sphere, we can calculate its volume using the formula for the volume of a sphere:
V_sphere = 4/3 * pi * r^3
Plugging in the value of r, we get:
V_sphere = 4/3 * pi * ((25 * c * m^3)^(1/3) / ∛2)^3
= 4/3 * pi * (25 * c * m^3) * (∛2/∛2)^3
= 4/3 * pi * 25 * c * m^3 * 2/∛4
= 4/3 * pi * 25/2 * c * m^3
= 100/6 * pi * c * m^3
= 50/3 * pi * c * m^3
So, the volume of the sphere is 50/3 * pi * c * m^3.
Therefore, the correct answer is option D: 50/3 * pi*c * m^3.