The diagram shows a solid metal cylinder.
The cylinder has base radius 2x and height 9x.
E cylinder is melted down and made into a sphere of radius r.
Find the expression for r in terms of x.
V = πr^2h = π(2x)^2(9x) = 4/3πr^2
This is volume of the cylinder and volume of the sphere. You can simplify this.
To find the expression for the radius of the sphere, we need to equate the volume of the cylinder with the volume of the sphere.
The volume of a cylinder is given by the formula V_cylinder = πr_cylinder^2h_cylinder, where r_cylinder is the radius of the base and h_cylinder is the height.
In this case, the base radius of the cylinder is given as 2x, and the height is given as 9x. Therefore, the volume of the cylinder is V_cylinder = π(2x)^2(9x) = 36πx^3.
The volume of a sphere is given by the formula V_sphere = (4/3)πr_sphere^3, where r_sphere is the radius of the sphere.
Since the cylinder has been melted down to form the sphere, the volume of the cylinder is equal to the volume of the sphere. Therefore, we can equate the volumes:
36πx^3 = (4/3)πr_sphere^3.
Now, let's solve for r_sphere, the radius of the sphere:
Divide both sides of the equation by π:
36x^3 = (4/3)r_sphere^3.
Multiply both sides of the equation by 3/4 to isolate r_sphere:
(3/4)(36x^3) = r_sphere^3.
Simplify:
27x^3 = r_sphere^3.
Now, take the cube root of both sides to solve for r_sphere:
r_sphere = (27x^3)^(1/3).
Simplify:
r_sphere = 3x.
Therefore, the expression for the radius of the sphere in terms of x is r = 3x.
To find the expression for the radius of the sphere (r) in terms of x, we need to use the relationship between the volume of the cylinder and the volume of the sphere.
1. Start with the volume of the cylinder formula:
V_cylinder = π * (base radius)^2 * height
2. Substitute the given values for the base radius and height of the cylinder:
V_cylinder = π * (2x)^2 * 9x
= π * 4x^2 * 9x
= 36πx^3
3. Since the cylinder is melted down and made into a sphere, the volume of the sphere will be equal to the volume of the cylinder:
V_sphere = V_cylinder
(4/3)π * r^3 = 36πx^3
4. Divide both sides of the equation by (4/3)π to isolate r^3:
r^3 = (36πx^3) / (4/3)π
= 27x^3
5. Take the cube root of both sides to solve for r:
r = ∛(27x^3)
Therefore, the expression for the radius of the sphere (r) in terms of x is ∛(27x^3).