Problem Breakdown:

We are given:

1. A triangle with a base of $10 \, \text{cm}$ and height $11 \, \text{cm}$.
2. A rectangle with a length of $22 \, \text{cm}$ and unknown width.
3. The area of the rectangle is $8$ times the area of the triangle.
4. We need to calculate the width of the rectangle.

Step 1: Calculate the Area of the Triangle

The formula for the area of a triangle is: $\text{Area of triangle} = \frac{1}{2} \times \text{base} \times \text{height}$ Substitute the given values: $\text{Area of triangle} = \frac{1}{2} \times 10 \times 11 = 55 \, \text{cm}^2$

Step 2: Calculate the Area of the Rectangle

Since the area of the rectangle is $8$ times the area of the triangle: $\text{Area of rectangle} = 8 \times \text{Area of triangle}$ $\text{Area of rectangle} = 8 \times 55 = 440 \, \text{cm}^2$

Step 3: Use the Rectangle's Area Formula

The formula for the area of a rectangle is: $\text{Area of rectangle} = \text{length} \times \text{width}$ Substitute the known values: $440 = 22 \times \text{width}$

Step 4: Solve for the Width

Rearrange the equation to find the width: $\text{width} = \frac{440}{22} = 20 \, \text{cm}$

Final Answer:

The width of the rectangle is: $\boxed{20 \, \text{cm}}$

Let me know if you'd like a detailed explanation of any step! Here are 5 related questions to expand your understanding:

1. How would the calculation change if the triangle's base or height were different?
2. What is the formula for the area of other shapes, such as circles or trapeziums?
3. How can you verify the result using proportions or estimation?
4. How would this problem change if the relationship between the areas were not a whole number (e.g., $8.5$ times)?
5. Can you derive the formula for the area of a triangle from scratch?

Tip: Always double-check the units and ensure consistency in area and dimensions during calculations!