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### Problem Description:
There are two tanks (Tank 1 and Tank 2) filling up with water at a constant rate. 
- **Tank 1** is a cylindrical tank.
- **Tank 2** is a cone-shaped tank. 

The **diameter (d)** of the surface of the water and the **height (h)** of the water are the two variables being considered.

The tasks are:
1. **Part a**: Determine which graph shows the relationship between diameter \( d \) and height \( h \) for **Tank 1**.
2. **Part b**: Sketch the graphs for Tank 2 showing:
   - i) The relationship between height \( h \) and time.
   - ii) The relationship between diameter \( d \) and height \( h \).

### Solutions:

#### Part a: Relationship between \( d \) and \( h \) for Tank 1 (Cylinder)
For **Tank 1** (a cylindrical tank), the diameter \( d \) remains constant as the water level rises. This means that the diameter is **independent of the height**. The correct graph would show a **horizontal line**, where \( d \) does not change as \( h \) increases.

- Looking at the graphs provided (A, B, C, and D), **Graph C** best represents this constant relationship (horizontal line) because \( d \) is constant regardless of the height \( h \).

#### Part b: Graphs for Tank 2 (Cone)

- **i) Relationship between \( h \) and time**:
  Since the cone is being filled at a constant rate, the height \( h \) will increase, but because the volume of a cone increases quadratically with height, the increase in height will **slow down** over time. The graph would be a **concave downward curve** (the rate of height increase slows as the tank fills).

- **ii) Relationship between \( d \) and \( h \)**:
  In a cone, the diameter \( d \) and the height \( h \) are proportional because the surface area of the water is increasing in size as the water level rises. The relationship between \( d \) and \( h \) will be **linear** because the diameter grows proportionally with height.

### Would you like to discuss these solutions further or have more details?

Here are five related questions:
1. How do you calculate the volume of water in a cone as it fills?
2. Why does the rate of height increase slow down in Tank 2 as it fills up?
3. How would the graph of \( h \) vs. time for Tank 1 (cylindrical tank) differ from that of Tank 2?
4. How would the graphs change if Tank 2 were an inverted cone?
5. How can we derive the equation for the proportional relationship between \( d \) and \( h \) for a conical tank?

**Tip**: When dealing with containers like cones and cylinders, understanding how the geometry affects volume helps in predicting how variables like height change over time!

Tanks 1 and 2 both start out empty and are filling up with water at a constant rate. The diameter of the surface of the water is d and the height of the water is h. Part a) Which of the graphs shows the relationship between d and h for tank 1? Part b) Sketch the graph for i) the relationship between h and time for tank 2, and ii) the relationship between d and h for tank 2.

Understand the relationship between diameter and height in cylindrical and conical tanks. Explore key geometry concepts and sketch graphs for the proportionality between the surface diameter and height for tanks filling at a constant rate. Suitable for Grades 8-10.

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