
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system

(a) Tank A and Tank B are rectangular prisms and are sitting on a flat table. Tank A has dimensions \(10 \text{ cm} \times 8 \text{ cm} \times 6 \text{ cm}\) and is sitting on one of its \(10 \text{ cm} \times 8 \text{ cm}\) faces. Tank B has dimensions \(5 \text{ cm} \times 9 \text{ cm} \times 8 \text{ cm}\) and is sitting on one of its \(5 \text{ cm} \times 9 \text{ cm}\) faces. Initially, Tank A is full of water and Tank B is empty. The water in Tank A drains out at a constant rate of \(4 \text{ cm}^3/\text{s}\). Tank B fills with water at a constant rate of \(4 \text{ cm}^3/\text{s}\). Tank A begins to drain and Tank B begins to fill at the same time.

(i) Determine after how many seconds Tank B will be exactly \(\frac{1}{3}\) full.

(ii) Determine the depth of the water left in Tank A at the instant when Tank B is full.

(iii) At one instant, the depth of the water in Tank A is equal to the depth of the water in Tank B. Determine this depth.

(b) Tank C is a rectangular prism with dimensions \(31 \text{ cm} \times 4 \text{ cm} \times 4 \text{ cm}\). Tank C sits on the flat table on one of its \(31 \text{ cm} \times 4 \text{ cm}\) faces. Tank D is in the shape of an inverted square-based pyramid. Its square base is parallel to the flat table and its fifth vertex touches the flat table. The height of Tank D is \(10 \text{ cm}\) and the side length of its square base is \(20 \text{ cm}\). Initially, Tank C is full of water and Tank D is empty. Tank D begins filling with water at a rate of \(1 \text{ cm}^3/\text{s}\). Two seconds after Tank D begins to fill, Tank C begins to drain at a rate of \(2 \text{ cm}^3/\text{s}\). At one instant, the volume of water in Tank C is equal to the volume of water in Tank D. Determine the depth of the water in Tank D at that instant.

Present the answer in LaTex format: \boxed{Your answer}
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