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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:
1. The surface area includes:
   - The base of the cylinder (\(\pi r^2\)).
   - The curved surface area (CSA), which is given by \(2\pi r h\), where \(h\) is the height of the water.

#### Given values:
- Radius (\(r\)) = 7 cm
- Height of the water (\(h\)) = 12 cm
- \(\pi = \frac{22}{7}\)

---

#### Step 1: Calculate the base area (\(\pi r^2\)):
\[
\pi r^2 = \frac{22}{7} \times 7 \times 7 = 154 \, \text{cm}^2
\]

#### Step 2: Calculate the curved surface area (\(2\pi r h\)):
\[
2\pi r h = 2 \times \frac{22}{7} \times 7 \times 12
\]
\[
2\pi r h = 2 \times 22 \times 12 = 528 \, \text{cm}^2
\]

#### Step 3: Total surface area:
\[
\text{Total Surface Area} = \pi r^2 + 2\pi r h
\]
\[
\text{Total Surface Area} = 154 + 528 = 682 \, \text{cm}^2
\]

---

### **Answer for (a):** \(682 \, \text{cm}^2\)

---

### **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:
\[
\text{Volume of cuboid} = \text{Length} \times \text{Width} \times \text{Height}
\]
\[
\text{Volume of cuboid} = 7 \times 8 \times 11 = 616 \, \text{cm}^3
\]

---

#### Volume of water in the cylinder after adding the cuboid:
The volume of the cylinder increases by \(616 \, \text{cm}^3\).

#### Step 1: Initial volume of the cylinder:
\[
\text{Initial Volume of Cylinder} = \pi r^2 h = \frac{22}{7} \times 7 \times 7 \times 12
\]
\[
\text{Initial Volume of Cylinder} = 22 \times 7 \times 12 = 1848 \, \text{cm}^3
\]

#### Step 2: New volume of the cylinder:
\[
\text{New Volume of Cylinder} = 1848 + 616 = 2464 \, \text{cm}^3
\]

---

#### Step 3: Calculate the new height of water:
The height of water (\(h\)) is given by:
\[
h = \frac{\text{Volume}}{\pi r^2}
\]
\[
h = \frac{2464}{\frac{22}{7} \times 7 \times 7}
\]
\[
h = \frac{2464}{154} = 16 \, \text{cm}
\]

---

### **Answer for (b):** The new height of water is \(16 \, \text{cm}\).

---

### **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:
\[
\text{Total Volume of Water} = 1848 \, \text{cm}^3
\]

---

#### Step 2: Convert to milliliters:
Since \(1 \, \text{cm}^3 = 1 \, \text{ml}\):
\[
\text{Total Volume of Water} = 1848 \, \text{ml}
\]

---

#### Step 3: Calculate the number of full bottles:
\[
\text{Number of full bottles} = \frac{\text{Total Volume of Water}}{\text{Capacity of each bottle}}
\]
\[
\text{Number of full bottles} = \frac{1848}{150} = 12.32
\]

Since only full bottles are counted:
\[
\text{Number of full bottles} = 12
\]

---

### **Answer for (c):** \(12\) full bottles

---

## Do you want further details or have any questions?

### Relative Questions:
1. How is the curved surface area of a cylinder derived?
2. How do we calculate the volume of irregular shapes submerged in water?
3. Why is the height of the water proportional to the total volume in the cylinder?
4. What would happen to the water level if the cuboid were larger?
5. How do we handle leftover liquid when filling bottles?

### Tip:
Always double-check unit conversions, especially between \(\text{cm}^3\) and \(\text{ml}\), to avoid calculation errors.

a) A cylindrical container of radius 7cm contains water to a depth of 12cm. Calculate the surface area of the container that is in contact with the water (π = 22/7). b) A cuboid measuring 7cm by 8cm by 11cm is put into the water in the cylinder. Find the new height of the water in the cylinder (π = 22/7). c) The water in (a) above was transferred into bottles of capacity 150ml. Find the number of full bottles obtained (π = 22/7).

Solve a multi-step geometry problem involving a cylindrical container and cuboid. Learn how to calculate surface area in contact with water, find the new water height after adding a cuboid, and determine the number of full bottles obtained. Includes key formulas and step-by-step solutions.

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