Final answer:
To find the height of each bottle, we first calculate the volume of the hemisphere bowl. Then, we calculate the volume of liquid wasted and subtract it from the total volume. Finally, we divide the remaining volume by the number of bottles to find the height of each bottle.
Step-by-step explanation:
To find the height of each bottle, we first need to find the volume of the hemisphere bowl and then calculate the volume of liquid wasted. We can then calculate the volume of liquid that is filled in each bottle and use it to find the height of the bottle.
The volume of a hemisphere is given by V = (2/3)πr³, where r is the radius. In this case, the diameter of the bowl is 36 cm, so the radius = 18 cm.
Volume of hemisphere = (2/3) × π × 18^3 = 24408 cm³
10% of liquid is wasted, so the volume wasted = 0.1 × 24408 cm³ = 2440.8 cm³
The remaining volume of liquid = Volume of hemisphere - Volume wasted = 24408 cm³ - 2440.8 cm³ = 21967.2 cm³
Now, we can find the volume of liquid in each bottle. The volume of a cylinder is given by V = πr²h, where r is the radius and h is the height.
In this case, the diameter of each bottle is 6 cm, so the radius = 3 cm.
Volume of each bottle = π × 3^2 × h = 9πh cm³
Since there are 72 bottles, the total volume of liquid = 72 × Volume of each bottle = 72 × 9πh cm³
Setting this equal to the remaining volume of liquid and solving for h:
72 × 9πh = 21967.2 cm³
h = 21967.2 cm³ / (72 × 9π)
h ≈ 32.12 cm