The volume of a cone is 25/3 π cm^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)
The volume of a cone is given by the formula V = (1/3)πr^2h, where r is the radius and h is the height.
Since the volume of the cone is 25/3 π cm^3, we have (1/3)πr^2h = 25/3 π.
Canceling out common terms of π and multiplying both sides by 3 gives r^2h = 25.
Now, since the height of the cone is equal to the sphere's diameter, we have h = 2r.
Substituting this into the equation r^2h = 25 gives r^2(2r) = 25.
Expanding the equation gives 2r^3 = 25.
Dividing both sides by 2 gives r^3 = (25/2).
Taking the cube root of both sides gives r = ((25/2)^(1/3)).
The volume of a sphere is given by the formula V = (4/3)πr^3.
Substituting the value of r we found earlier, we get V = (4/3)π((25/2)^(1/3))^3.
Simplifying this expression gives V = (4/3)π(125/8).
Multiplying and dividing gives V = (500/24)π.
Therefore, the volume of the sphere is (500/24)π cm^3. Answer: \boxed{\frac{500}{24}\pi}.
wrong
Apologies for the mistake. Let's solve the problem correctly.
The volume of a cone is given by the formula V_cone = (1/3)πr^2h, where r is the radius and h is the height.
The volume of the cone is given as 25/3π cm^3. Therefore, we have:
(1/3)πr^2h = 25/3π
Canceling out common terms of π, we get:
r^2h = 25
Since the height of the cone is equal to the sphere's diameter, we have h = 2r.
Substituting this into the equation r^2h = 25, we get:
r^2(2r) = 25
Expanding and rearranging the equation gives:
2r^3 = 25
Dividing both sides by 2 gives:
r^3 = 25/2
Taking the cube root of both sides gives:
r = (25/2)^(1/3)
Now, let's find the volume of the sphere. The volume of a sphere is given by the formula V_sphere = (4/3)πr^3.
Substituting the value of r we found earlier, we get:
V_sphere = (4/3)π((25/2)^(1/3))^3
Simplifying this expression gives:
V_sphere = (4/3)π(125/8)
Multiplying and simplifying further gives:
V_sphere = (500/24)π
Therefore, the volume of the sphere is 500/24π cm^3. Answer: 500/24π.
To find the volume of the sphere, we need to first determine the radius of the cone.
Given that the volume of the cone is 25/3 π cm^3, we can use the formula for the volume of a cone:
V = 1/3πr^2h,
where V is the volume, r is the radius, and h is the height.
Since the volume of the cone is already given as 25/3 π cm^3, and the height is equal to the sphere's diameter, we have:
25/3 π = 1/3πr^2(2r)
= 2/3πr^3.
Now, we can solve for the radius, r:
25/3 = 2/3r^3
25 = 2r^3
r^3 = 25/2
r = (25/2)^(1/3).
With the radius of the cone now known, we can proceed to find the volume of the sphere.
The formula for the volume of a sphere is:
V = (4/3)πr^3,
where V is the volume and r is the radius.
Substituting the radius obtained from the cone into the formula, we get:
V = (4/3)π((25/2)^(1/3))^3.
Simplifying this expression gives us the volume of the sphere.