A cylindrical tin 8cm in diameter contains water to a depth of 4cm. if a cylindrical wooden rod 4cm in diameter and 6cm long is placed in the tin it floats exactly half submerged. what is the New depth of water

To find the new depth of water when the wooden rod is placed in the tin, we need to consider the volume of the wooden rod and how it displaces water.

Let's find the volume of water that was displaced by the rod:
1. The water initially filled up to a depth of 4 cm.
2. The diameter of the tin is 8 cm, so the radius is 8/2 = 4 cm.
3. The initial volume of water in the tin can be calculated using the formula for the volume of a cylinder: V = πr^2h.
- Substituting the values: V = π * 4^2 * 4 = 64π cm^3 (approximately 201.06 cm^3).

Now, let's find the volume of the wooden rod that floats half-submerged:
1. The diameter of the wooden rod is 4 cm, so the radius is 4/2 = 2 cm.
2. The length of the rod is 6 cm.
3. The volume of the rod can be calculated using the formula for the volume of a cylinder: V = πr^2h.
- Substituting the values: V = π * 2^2 * 6 = 24π cm^3 (approximately 75.4 cm^3).

When the rod is added, it displaces an amount of water equal to its own volume.

To find the new depth of water, we need to determine the volume that remains in the tin after adding the rod:
1. Subtract the volume of the rod from the initial volume of water: 201.06 cm^3 - 75.4 cm^3 = 125.66 cm^3.

Now, let's find the new depth of water using the remaining volume:
1. The radius of the tin remains the same at 4 cm.
2. The volume can be calculated using the formula for the volume of a cylinder: V = π * r^2 * h.
- Rearrange the formula to solve for h: h = V / (π * r^2).
- Substituting the values: h = 125.66 cm^3 / (π * 4^2) ≈ 1.991 cm.

Therefore, the new depth of water in the tin, with the wooden rod placed in it, is approximately 1.991 cm.

To solve this problem, we need to consider the volume of the water displaced by the wooden rod and the resulting change in the water level. Here's how we can approach it:

1. Calculate the volume of the water initially in the tin using the formula for the volume of a cylinder:
Volume = π * (radius^2) * height

The radius of the tin is half of its diameter, so the initial volume is:
Volume_initial = π * (4cm/2)^2 * 4cm = 32π cm^3

2. Determine the volume of the wooden rod using the same formula:
Volume_wooden_rod = π * (radius^2) * height
Volume_wooden_rod = π * (2cm/2)^2 * 6cm = 6π cm^3

3. The volume of water displaced by the wooden rod is equal to the volume of the rod itself since it is floating half-submerged. So, the displaced volume of water is:
Volume_displaced = Volume_wooden_rod = 6π cm^3

4. When the wooden rod is placed in the tin, it displaces an equal volume of water, causing the water level to rise.

5. To find the new depth, we need to find the new height of water in the tin.

Subtract the volume of the wooden rod from the initial volume of water and divide it by the cross-sectional area (π * (radius^2)) to obtain the new height:
New_height = (Volume_initial - Volume_displaced) / (π * (radius^2))
New_height = (32π cm^3 - 6π cm^3) / (π * (4cm/2)^2)
New_height = 26π cm^3 / (π * 2^2) cm^2
New_height = 26π cm^3 / (4π) cm^2 = 6.5 cm

Therefore, the new depth of water in the tin is 6.5 cm.

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