A hollow cylinder has solid hemisphere in one end (bottom) and on the other end (top), it is open. The cylinder is filled with water upto the height 10 cm. Find the volume of water if the common diameter is 8 cm. [Answer in rational form]. Also, if the top end is covered with a plate and it is reversed, then find height of the water now.
@mathhelper
To calculate h ,
Do we need to subtract volume of cylinder - volume of hemisphere
Or your way is right?????
To find the volume of water in the cylinder, we need to determine the volume of the hollow cylinder and subtract the volume of the solid hemisphere.
1. Volume of the hollow cylinder:
The formula for the volume of a hollow cylinder is given by V1 = πr^2h1, where r is the radius and h1 is the height of the hollow cylinder.
Given that the radius of the cylinder is half the diameter, which is 8 cm, so the radius (r) = 8/2 = 4 cm.
The height of the hollow cylinder (h1) is the total height of the water, which is 10 cm.
Substituting the values into the formula, we have V1 = π(4 cm)^2 * 10 cm.
2. Volume of the solid hemisphere:
The formula for the volume of a solid hemisphere is given by V2 = (2/3)πr^3, where r is the radius of the hemisphere.
Given that the radius of the hemisphere is half the diameter, which is 8 cm, so the radius (r) = 8/2 = 4 cm.
Substituting the value into the formula, we have V2 = (2/3)π(4 cm)^3.
3. Total volume of water:
The volume of water is the difference between the volume of the hollow cylinder and the volume of the hemisphere, so V_water = V1 - V2.
Substituting the values we calculated earlier, V_water = π(4 cm)^2 * 10 cm - (2/3)π(4 cm)^3.
Simplifying the equation, V_water = 160π - (2/3)(64π).
Now, to find the height of the water when the top end is covered with a plate and reversed, we need to consider the volume of the hemisphere and the volume of the water.
4. Volume of the hemisphere:
We already calculated the volume of the hemisphere as (2/3)π(4 cm)^3.
5. Volume of water with covered top end:
When the top end is covered and reversed, the water fills the hemisphere. Therefore, the volume of water is equal to the volume of the hemisphere. So, V_water_reversed = V2.
Substituting the value of V2 calculated earlier, V_water_reversed = (2/3)π(4 cm)^3.
To summarize:
- The volume of water with open top end = 160π - (2/3)(64π).
- The volume of water with covered top end = (2/3)π(4 cm)^3.
I hope this explanation helps!
ambigious:
1. "upto the height 10 cm"
does the height of 10 cm include the height of the hemisphere, or is it
10 cm above the hemisphere part ?
2. If the hemisphere is solid, it can't hold any water, I will assume it holds water
I will conclude it is 4 cm of hemisphere + 6 cm of cylinder
Vol = (1/2)(4/3)π (4^3) + π(4^2)(6) = 416/3 π cm^3
If turned upside down:
416/3 π = π(4^2)(h)
416/3 = 16h
h = 26/3
the water height would be 26/3 cm
Make corrections to your post if I did not interpret it correctly.