Math Problem Statement
Solution
Let's work through the problem step by step:
a) Surface area of the cylindrical container in contact with the water
Formula for the surface area in contact with the water:
- The surface area includes:
- The base of the cylinder ().
- The curved surface area (CSA), which is given by , where is the height of the water.
Given values:
- Radius () = 7 cm
- Height of the water () = 12 cm
Step 1: Calculate the base area ():
Step 2: Calculate the curved surface area ():
Step 3: Total surface area:
Answer for (a):
b) New height of water in the cylinder after adding the cuboid
Volume of water displaced by the cuboid:
The volume of the cuboid is calculated using the formula:
Volume of water in the cylinder after adding the cuboid:
The volume of the cylinder increases by .
Step 1: Initial volume of the cylinder:
Step 2: New volume of the cylinder:
Step 3: Calculate the new height of water:
The height of water () is given by:
Answer for (b): The new height of water is .
c) Number of full bottles of capacity 150 ml
Step 1: Total volume of water:
The total volume of water is the volume of the cylinder:
Step 2: Convert to milliliters:
Since :
Step 3: Calculate the number of full bottles:
Since only full bottles are counted:
Answer for (c): full bottles
Do you want further details or have any questions?
Relative Questions:
- How is the curved surface area of a cylinder derived?
- How do we calculate the volume of irregular shapes submerged in water?
- Why is the height of the water proportional to the total volume in the cylinder?
- What would happen to the water level if the cuboid were larger?
- How do we handle leftover liquid when filling bottles?
Tip:
Always double-check unit conversions, especially between and , to avoid calculation errors.
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Math Problem Analysis
Mathematical Concepts
Geometry
Surface Area of Cylinders
Volume of Cylinders
Volume of Cuboids
Unit Conversion (cm³ to ml)
Formulas
Surface area of cylinder = πr² + 2πrh
Volume of cuboid = Length × Width × Height
Volume of cylinder = πr²h
Height of cylinder = Volume / (πr²)
Theorems
Volume displacement theorem
Proportionality of height to volume in cylinders
Suitable Grade Level
Grades 8-10