Suppose a sphere, cylinder, and cone all share the same radius. The cylinder and cone also share the same height, which is twice the radius. Which of the following is true about the relationship between the volume formulas for the sphere, cylinder, and cone?(1 point) Responses sphere = cylinder + cone sphere = cylinder + cone cylinder = sphere – cone cylinder = sphere – cone cone = sphere – cylinder cone = sphere – cylinder cone = cylinder – sphere
The correct relationship between the volume formulas for the sphere, cylinder, and cone is:
sphere = cylinder + cone
Use the relationship among the formulas for the volumes of cones, cylinders, and spheres to solve the following problem. The volume of a cylinder is 36 cm3 . What is the volume of a sphere if its radius is the same as the cylinder’s and the height of the cylinder is equal to the sphere’s diameter?(1 point) Responses 18 cm3 18 cm cubed 24 cm3 24 cm cubed 12 cm3 12 cm cubed 54 cm3
The volume of a cylinder is given by the formula V = πr^2h, where r is the radius and h is the height. In this case, we are given that the volume of the cylinder is 36 cm^3.
Let's call the radius of the cylinder and sphere as r. We are also given that the height of the cylinder is equal to the diameter of the sphere, so the height of the cylinder is 2r.
Using the relationship between the volume formulas for the cylinder and sphere, which is sphere = cylinder + cone, we can substitute the given values into the formula:
sphere = cylinder + cone
sphere = 36 + cone
Since the radius and height of the cone are the same as the cylinder, the volume of the cone can be expressed as 1/3 times the volume of the cylinder:
cone = (1/3)(πr^2)(2r)
cone = (2/3)(πr^3)
Substituting this into the equation for the volume of the sphere:
sphere = 36 + (2/3)(πr^3)
Now, we can solve for the volume of the sphere by substituting the given value of the volume of the cylinder (36 cm^3) and solving for πr^3:
sphere = 36 + (2/3)(πr^3)
πr^3 = sphere - 36
r^3 = (sphere - 36)/(π)
r = (sphere - 36)^(1/3)/(π^(1/3)) --(1)
We can also use the formula for the volume of the sphere, V = (4/3)πr^3, and substitute the given values to solve for the volume of the sphere:
(4/3)πr^3 = sphere
r^3 = (3/4)sphere/π
r = (3/(4π))^(1/3)sphere^(1/3) --(2)
Since both (1) and (2) represent the same radius, we can set them equal to each other and solve for the volume of the sphere:
(sphere - 36)^(1/3)/(π^(1/3)) = (3/(4π))^(1/3)sphere^(1/3)
Cubing both sides of the equation:
(sphere - 36)/(π) = (27/(64π))sphere
Rearranging the equation to isolate the variable:
sphere - 36 = (27/64)sphere
Simplifying:
(64/64)sphere - (27/64)sphere = 36
(37/64)sphere = 36
Multiplying both sides by (64/37):
sphere = (64/37) * 36
sphere ≈ 62.38 cm^3
Therefore, the volume of the sphere is approximately 62.38 cm^3.
that is not an option.
Use the relationship among the formulas for the volumes of cones, cylinders, and spheres to solve the following problem. The volume of a cylinder is 36 cm3 . What is the volume of a sphere if its radius is the same as the cylinder’s and the height of the cylinder is equal to the sphere’s diameter?(1 point) Responses 18 cm3 18 cm cubed 24 cm3 24 cm cubed 12 cm3 12 cm cubed 54 cm3
Apologies for the confusion. Let's solve the problem correctly.
The volume of a cylinder is given by the formula V = πr^2h, where r is the radius and h is the height/diameter of the cylinder. In this case, we are given that the volume of the cylinder is 36 cm^3.
Let's call the radius of the cylinder and sphere as r. We are also given that the height of the cylinder is equal to the diameter of the sphere, so the height of the cylinder is 2r.
Using the relationship between the volume formulas for the cylinder and sphere, which is sphere = cylinder + cone, we can substitute the given values into the formula:
sphere = cylinder + cone
sphere = 36 + cone
Since the radius and height of the cone are the same as the cylinder, the volume of the cone can be expressed as 1/3 times the volume of the cylinder:
cone = (1/3)(πr^2)(2r)
cone = (2/3)(πr^3)
Substituting this into the equation for the volume of the sphere:
sphere = 36 + (2/3)(πr^3)
Now, let's substitute the value of the given cylinder volume (36 cm^3) into the equation:
sphere = 36 + (2/3)(πr^3)
We are trying to find the volume of the sphere, so we need to isolate the sphere. Let's solve for r:
(2/3)(πr^3) = sphere - 36
(2/3)(πr^3) = sphere - 36
Now, let's solve for the volume of the sphere by substituting the expression for (πr^3):
(2/3)(πr^3) = sphere - 36
(2/3)(πr^3) = sphere - 36
(2/3)(πr^3) = sphere - 36
Multiplying both sides by 3/(2π):
r^3 = (sphere - 36)(3/(2π))
r^3 = (3(sphere - 36))/(2π)
Taking the cube root of both sides:
r = (3(sphere - 36))/(2π)^(1/3)
Since the radius of the sphere is the same as the radius of the cylinder, r = 36^(1/2) = 6.
Substituting r = 6 into the equation:
6 = (3(sphere - 36))/(2π)^(1/3)
Multiplying both sides by (2π)^(1/3)/3:
(2π)^(1/3) = sphere - 36
Adding 36 to both sides:
sphere = (2π)^(1/3) + 36
Approximating the value of (2π)^(1/3) as 1.395, we have:
sphere ≈ 1.395 + 36
sphere ≈ 37.395 cm^3
Therefore, the volume of the sphere is approximately 37.395 cm^3.
answer the question using only the answer choices provided.
Apologies for the misunderstanding. Let's solve the problem using the given answer choices.
The volume of the cylinder is 36 cm^3. We know that the volume of a cylinder is given by the formula V = πr^2h, where r is the radius and h is the height/diameter of the cylinder.
Let's call the radius of the cylinder and sphere as r. Since the height of the cylinder is twice the radius, we have h = 2r.
Substituting the values into the volume formula for the cylinder:
36 = πr^2(2r)
Simplifying:
36 = 2πr^3
Dividing both sides by 2π:
r^3 = 18/π
Taking the cube root of both sides:
r = (18/π)^(1/3)
Now, to find the volume of the sphere, we can use the formula V = (4/3)πr^3.
Plugging in the value of r:
V = (4/3)π[(18/π)^(1/3)]^3
V = (4/3)π(18/π)
V = 24 cm^3
Therefore, the volume of the sphere is 24 cm^3, which matches with one of the provided answer choices.
The volume of a cone is 253 π cm3 . 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) Responses 256 π cm3 Start Fraction 25 over 6 End Fraction pi cm cubed 252 π cm3 Start Fraction 25 over 2 End Fraction pi cm cubed 25 π cm3 25 pi cm cubed 503 π cm3
The volume of the cone is given as 253π cm^3.
Let's call the radius of the cone and sphere as r. We are given that the height of the cone is equal to the diameter of the sphere, so the height of the cone is 2r.
The volume of a cone is given by the formula V = 1/3πr^2h, where r is the radius and h is the height.
Substituting the given values into the volume formula for the cone:
253π = 1/3πr^2(2r)
Simplifying:
253 = 2/3r^3
Multiplying both sides by 3/2:
379.5 = r^3
Taking the cube root of both sides:
r = (379.5)^(1/3)
Now, to find the volume of the sphere, we can use the formula V = 4/3πr^3.
Plugging in the value of r:
V = 4/3π[(379.5)^(1/3)]^3
V = 4/3π(379.5)
V = 1598π/3 cm^3
Therefore, the volume of the sphere is approximately 532.67π cm^3, which is not one of the provided answer choices. It seems that the given answer choices may not be correct or there is an error in the question.