What length of a solid cylinder 2cm in diameter must be taken to recast into hollow cylinder of length 16cm,external diameter 2cm and thickness 2.5mm?

What length of solid cylinder 2 cm in diameter must be taken to recast into a hollow cylinder of length 16 cm x 10 external diameter 20 cm and thickness 2.5 mm

To find the length of the solid cylinder needed to recast it into a hollow cylinder with given dimensions, we need to calculate the volume of the solid cylinder and subtract the volume of the hollow cylinder.

Step 1: Calculate the volume of the solid cylinder.
The volume of a solid cylinder is given by the formula:
V_solid = π * r^2 * h_solid
where r is the radius of the solid cylinder (which is half of the diameter) and h_solid is the height or length of the solid cylinder.

The diameter of the solid cylinder is given as 2 cm, so the radius (r) is 2 cm / 2 = 1 cm = 0.01 meters.

Step 2: Calculate the volume of the hollow cylinder.
The volume of a hollow cylinder is given by the formula:
V_hollow = π * (R^2 - r^2) * h_hollow
where R is the external radius of the hollow cylinder, r is the internal radius of the hollow cylinder, and h_hollow is the height or length of the hollow cylinder.

The external diameter of the hollow cylinder is given as 2 cm, so the external radius (R) is 2 cm / 2 = 1 cm = 0.01 meters.

The thickness of the hollow cylinder is given as 2.5 mm, so the internal radius (r) is (2 cm - 2 * 2.5 mm) / 2 = (20 mm - 5 mm) / 2 = 7.5 mm / 2 = 3.75 mm = 0.0375 cm = 0.000375 meters.

The length of the hollow cylinder (h_hollow) is given as 16 cm = 0.16 meters.

Step 3: Calculate the difference in volume.
V_diff = V_solid - V_hollow

Step 4: Substitute the values into the formulas and calculate.
V_solid = π * (0.01 meters)^2 * h_solid
V_hollow = π * ((0.01 meters)^2 - (0.000375 meters)^2) * 0.16 meters

V_diff = π * (0.01 meters)^2 * h_solid - π * ((0.01 meters)^2 - (0.000375 meters)^2) * 0.16 meters

Simplifying:
V_diff = π * 0.0001 meters^2 * h_solid - π * (0.0001 meters^2 - 0.000000140625 meters^2) * 0.16 meters

V_diff = 0.0001 * π * h_solid - 0.000000040625 * π * 0.16 meters^3

V_diff = 0.00001 * π * h_solid - 0.0000000065 * π meters^3

Step 5: Solve for h_solid.
To find the length of the solid cylinder, we need to equate the volume difference to zero and solve for h_solid.

0.00001 * π * h_solid - 0.0000000065 * π = 0

0.00001 * π * h_solid = 0.0000000065 * π

h_solid = 0.0000000065 * π / 0.00001 * π

h_solid = (0.0000000065 / 0.00001) * π

h_solid = 0.00065 * π

h_solid ≈ 0.0020432 meters or 0.20432 cm

Therefore, a solid cylinder of approximately 0.20432 cm in height or length needs to be taken to recast it into a hollow cylinder with the given dimensions.

for the hollow cylinder,

outside volume is pi * 1^2 * 16 = 16pi
inside volume = pi * (1-.25)^2 * 16 = 9pi

volume of metal = 7pi cm^3

volume of solid is thus

7pi = pi * 1^2 * h = h*pi
h = 7 cm