Suppose a sphere, cylinder, and cone all share the same radius and 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
1 cylinder = 2 cones + 1 sphere
1 cylinder = 2 cones + 1 sphere
3 cones = 1 cylinder
3 cones = 1 cylinder
sphere = cylinder + cone
sphere = cylinder + cone
sphere - cone = cylinder
The correct answer is: sphere = cylinder + cone
The volume of a sphere is 48 m3 . What is the volume of a cylinder if its radius is the same as the sphere’s and its height is equal to the sphere’s diameter?(1 point)
The volume of a sphere is given by the formula V = (4/3)πr^3, where r is the radius.
In this case, the volume of the sphere is 48 m^3.
So we can write the equation as:
48 = (4/3)πr^3
To find the radius, we need to isolate it.
Divide both sides of the equation by (4/3)π:
48 / ((4/3)π) = r^3
The (4/3) and π cancel on the right side, giving:
r^3 = (48 / ((4/3)π))
Multiply both sides by ((4/3)π) to solve for r^3:
r^3 = 48 * ((4/3)π)
Simplify:
r^3 = 64π
Take the cube root of both sides to find the radius:
r = (64π)^(1/3)
Now we can find the volume of the cylinder.
The formula for the volume of a cylinder is V = πr^2h, where r is the radius and h is the height.
In this case, the radius is the same as the sphere's radius and the height is equal to the sphere's diameter, which is twice the radius.
So the height of the cylinder is 2r.
Substitute these values into the formula:
V = π(r^2)(2r)
Simplify:
V = 2πr^3
Now we can substitute the value of r that we found earlier:
V = 2π((64π)^(1/3))^3
Evaluate the expression to find the volume of the cylinder.
what is the answer though?
To find the volume of the cylinder, we need to substitute the value of r that we found earlier into the formula:
V = 2π((64π)^(1/3))^3
Calculating this expression, we get:
V = 2π * (64π)^(3/3)
Simplifying further:
V = 2π * (64π)
V = 128π^2
So, the volume of the cylinder is 128π^2 cubic units.
The volume of a sphere is 48 m3. What is the volume of a cylinder if its radius is the same as the sphere’s and its height is equal to the sphere’s diameter?(1 point)
___m3
To find the volume of the cylinder, we need to determine the radius of the sphere first.
Given that the volume of the sphere is 48 m^3, we can use its formula:
V = (4/3)πr³
We rearrange the equation to solve for r:
r³ = (3V) / (4π)
Substituting the given volume of the sphere, we have:
r³ = (3(48)) / (4π)
r³ = 36 / π
Taking the cube root of both sides, we find:
r ≈ 3.061 m
Since the radius of the cylinder is the same as the sphere, the radius of the cylinder is also approximately 3.061 m.
The height of the cylinder is equal to the diameter of the sphere, which means it is twice the radius:
Height of cylinder ≈ 2 * 3.061 m = 6.122 m
Now, to find the volume of the cylinder, we can use its formula:
V = πr²h
Substituting the values we found:
V = π(3.061)²(6.122)
V ≈ π * 9.370281 * 6.122
V ≈ 179.95758 m³
Therefore, the volume of the cylinder is approximately 179.95758 m³.