This diagram shows a solid metal cylinder.
The cylinder hasbase radius 2x and height 9x.
The cylinder is melted down and made into a sphere of radius r
Find an expression for in terms of x
Volume of cylinder = ( 2 x ) ² ∙ π ∙ 9 x = 4 x² ∙ π ∙ 9 x = 36 x³ ∙ π
Volume of a sphere with radius r = ( 4 / 3 ) ∙ π ∙ r³
Volume of cylinder = Volume of a sphere
36 x³ ∙ π = ( 4 / 3 ) ∙ π ∙ r³
Divide both sides by π
36 x³ = ( 4 / 3 ) ∙ r³
Multiply both sides by 3
36 x³ ∙ 3 = 4 ∙ r³
108 x³ = 4 ∙ r³
Divide both sides by 4
27 x³ = r³
r³ = 27 x³
r = ∛ (27 x³ )
r = 3 x
Oh, the poor cylindrical metal met a melting pot of fate and transformed into a spherical shape! How dramatic! Now, let's find out the radius of the new sphere in terms of x.
To do that, we need to remember that the volume of the cylinder is equal to the volume of the sphere. Volume, volume, what a magical word!
The volume of the cylinder is given by the formula V_cylinder = πr_cylinder^2h_cylinder, where r_cylinder is the base radius and h_cylinder is the height.
So, for our cylinder, V_cylinder = π(2x)^2(9x) = 36πx^3.
Now, let's move on to the sphere. The volume of a sphere is V_sphere = (4/3)πr_sphere^3. Fun fact: when you roll a sphere down a hill, it just keeps on going and going!
Since the metal from the cylinder is used to make the sphere, the volumes are equal: 36πx^3 = (4/3)πr_sphere^3.
Now, we just need to solve for r_sphere. Let's punch some numbers.
36πx^3 = (4/3)πr_sphere^3.
Divide both sides by π to get rid of the pi's. We don't want pizza here!
36x^3 = (4/3)r_sphere^3.
To solve for r_sphere, we simply take the cube root of both sides:
r_sphere = (3(36x^3)/4)^(1/3).
Now, let's simplify this expression, because I know math is already complicated enough.
r_sphere = (108x^3/4)^(1/3).
And that, my friend, is your expression for the radius of the spherical creation out of our melted metal cylinder! Keep on rolling with that math!
To find the expression for the radius of the sphere in terms of x, we need to relate the volume of the cylinder to the volume of the sphere.
Given:
Base radius of the cylinder, r = 2x
Height of the cylinder, h = 9x
The volume of a cylinder is given by the formula V_cylinder = π * r^2 * h.
Substituting the given values into the formula, we get:
V_cylinder = π * (2x)^2 * (9x)
V_cylinder = π * 4x^2 * 9x
V_cylinder = 36πx^3
The volume of a sphere is given by the formula V_sphere = (4/3) * π * r^3.
Since both the cylinder and the sphere are made of the same material, their volumes should be equal. Therefore:
V_cylinder = V_sphere
Substitute the expression for the volume of the cylinder into the equation:
36πx^3 = (4/3) * π * r^3
Now, let's isolate r to find an expression for it.
Multiply both sides of the equation by (3/4) to simplify:
r^3 = (3/4) * 36x^3
Simplify further:
r^3 = 27x^3
Take the cube root of both sides to find the expression for r:
r = (27x^3)^1/3
Simplifying the cube root of 27:
r = 3x
Therefore, the expression for the radius of the sphere in terms of x is r = 3x.
To find an expression for the radius of the sphere in terms of x, we need to set up an equation by equating the volumes of the cylinder and the sphere.
The volume of the cylinder can be calculated using the formula V_cylinder = πr^2h, where r is the base radius and h is the height.
In this case, the base radius of the cylinder is 2x and the height is 9x. So, the volume of the cylinder is:
V_cylinder = π(2x)^2(9x)
= π(4x^2)(9x)
= 36πx^3
The volume of a sphere can be calculated using the formula V_sphere = (4/3)πr^3, where r is the radius of the sphere.
In this case, we want to find the radius of the sphere, so we can rearrange the formula as:
r = [3V_sphere / (4π)]^(1/3)
Now, we can substitute the volume of the cylinder into the formula for the radius of the sphere:
r = [3(36πx^3) / (4π)]^(1/3)
= (108x^3 / 4)^(1/3)
= (27x^3 / 4)^(1/3)
Therefore, the expression for the radius of the sphere in terms of x is (27x^3 / 4)^(1/3).