Two solid cylinders of equal diameter have different heights. They are...
Explanation:
To solve this problem, let's consider the initial and final dimensions of the two cylinders.
Let:
- D = diameter of the cylinders
- h1 = initial height of the shorter cylinder
- h2 = initial height of the longer cylinder
- h1' = final height of the shorter cylinder
- h2' = final height of the longer cylinder
Since both cylinders are compressed plastically by the same percentage reduction in their heights, we can set up the following equation:
(h1 - h1') / h1 = (h2 - h2') / h2
Step 1: Finding the final height of the longer cylinder
Since both cylinders have the same initial diameter, the percentage reduction in height is the same for both. Therefore, we can write:
(h1 - h1') / h1 = (h2 - h2') / h2
Simplifying this equation, we get:
h1' / h1 = h2' / h2
Cross-multiplying, we get:
h1' * h2 = h1 * h2'
Step 2: Finding the ratio of the final diameters
The initial and final volumes of the cylinders are the same since the material is incompressible. Therefore, we can write:
π * (D/2)^2 * h1 = π * (D/2)^2 * h1'
Cancelling out the common terms, we get:
h1 = h1'
Similarly, we can write:
π * (D/2)^2 * h2 = π * (D/2 * r)^2 * h2'
Cancelling out the common terms and simplifying, we get:
(D/2)^2 * h2 = (D/2 * r)^2 * h2'
D^2 * h2 = (D * r)^2 * h2'
Dividing both sides by h2 and taking the square root, we get:
D = D * r
Simplifying, we get:
1 = r
Therefore, the ratio of the final diameter of the shorter cylinder to that of the longer cylinder is 1.