a solid aluminum cube measuring 15 inches on a side is melted and rolled into an aluminum wire tat is 0.2 inches in diameter. what will be the length of this wire?
volume of original cube = 15^3 cubic inches
= 3375 cubic inches
The volume should stay the same.
Think of the wire as a "long" cylinder.
πr^2 h = 3375
π(.1^2)h = 3375
h = 3375/(.01π) = appr. 107430 inches
or appr. 8952 ft
To find the length of the wire formed from the melted aluminum cube, we need to consider the conservation of volume during the transformation.
First, let's calculate the volume of the aluminum cube:
Volume of the cube = (Side length)^3
Volume of the cube = (15 inches)^3 = 3375 cubic inches
Next, let's calculate the volume of the aluminum wire. Since the wire is in a cylindrical form, we will use the formula for the volume of a cylinder:
Volume of the cylinder = π * (radius)^2 * height
Given that the diameter of the wire is 0.2 inches, we can calculate the radius by halving the diameter:
Radius = 0.2 inches / 2 = 0.1 inches
Now let's calculate the height of the cylinder, which is the length of the wire we are looking for.
Using the formula for volume, we know that the volume of the cube and the volume of the cylinder are equal:
3375 cubic inches = π * (0.1 inches)^2 * height
Simplifying:
3375 = 0.01π * height
height = 3375 / (0.01π)
height ≈ 107525.37 inches
Therefore, the length of the wire will be approximately 107525.37 inches.