"The diagram shows a solid cylinder. The cylinder has a radius of 2y and height of 6y.
The cylinder is melted down and made into a sphere of radius r.
Express r in terms of y."
this is due tmrw and im on the verge of a breakdown 💀
volume of cylinder:
Area of circle • height =
( 2 y)^2π • 6 y = 4 y^2 π • 6 y =
24 y^3 π
When the cylinder is melted down volume of material is same.
volume of sphere:
4 / 3 r^3 π
24 y^3 π = 4 / 3 r^3 π
Divide both dides by 4 π
6 y^3 = r^3 / 3
Multiply both sided by 3
18 y^3 = r^3
r^3 = 18 y^3
r = ∛18 y
go with Bosnian. My typo didn't include the y.
for a cylinder,
v = πr^2 h = π(2y)^2 * 6y = 24πy^3
for a sphere,
v = 4/3 πr^3 = 24πy^3
so r^3 = 3/(4Ï€) * 24Ï€y^3 = 18
r = ∛18
tysm!! i got 100% :))
Well, it seems like we have a melting situation here! Let's see if we can turn this cylinder into a sphere, shall we?
Now, 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 of the cylinder is 2y and the height is 6y. So, the volume of the cylinder is V = π(2y)^2(6y).
But wait! We want to transform this cylinder into a sphere, so we need to find the radius of the sphere. Lucky for us, the volume of a sphere is V = (4/3)Ï€r^3.
Since the volume of the cylinder is equal to the volume of the sphere, we can set the two formulas equal to each other:
Ï€(2y)^2(6y) = (4/3)Ï€r^3
Now, let's simplify this equation. First, cancel out the π's:
(2y)^2(6y) = (4/3)r^3
Expanding and simplifying:
4y^2(6y) = (4/3)r^3
Now, divide both sides by 4y^2 to isolate r:
6y = (4/3)r^3 / 4y^2
Simplifying again:
6y = (1/3)r^3 / y^2
Multiplying both sides by y^2:
6y^3 = (1/3)r^3
Finally, we can take the cube root of both sides to find the value of r:
∛(6y^3) = ∛((1/3)r^3)
r = ∛(18y^3)
So, in terms of y, the radius of the sphere is r = ∛(18y^3).
I hope that wasn't too "melting" for you!
To find the value of r in terms of y, we need to relate the properties of the cylinder to those of the sphere.
Let's start by looking at the properties of the cylinder. We are given that the cylinder has a radius of 2y and a height of 6y. The volume of a cylinder is given by the formula V = πr^2h, where r is the radius and h is the height.
Substituting the given values into the formula, we have V = π(2y)^2 * 6y = 24πy^3.
Since we know that the cylinder is melted down and made into a sphere, the volume of the sphere must be the same as the volume of the cylinder. The volume of a sphere is given by the formula V = (4/3)Ï€r^3, where r is the radius.
Set the cylinder volume equal to the sphere volume and solve for r:
24Ï€y^3 = (4/3)Ï€r^3
Simplifying the equation, we can cancel out π and divide both sides by (4/3):
6y^3 = r^3
To get r, we take the cube root of both sides:
r = (6y^3)^(1/3)
Simplifying the cube root of 6y^3, we have:
r = 2y^(1/3)
Therefore, the radius of the sphere, r, is equal to 2y raised to the power of one-third.