A scale model of an industrial complex has a scale of 1 inch:22 ft. The roof on the model measures 162 square inches. Find the area of the actual roof. Round to the nearest square foot.

To find the area of the actual roof based on the scale of the model, follow these steps:

### Step 1: Understand the scale
The scale is given as \( 1 \, \text{inch} : 22 \, \text{feet} \). This means that 1 inch on the model represents 22 feet in real life.

### Step 2: Relate the areas
Since the model is two-dimensional (the roof is a surface), the area will scale according to the square of the linear scale factor. The linear scale factor is 22, so the area scale factor is \( 22^2 \).

\[
\text{Area Scale Factor} = 22^2 = 484
\]

### Step 3: Calculate the actual area
The area of the roof on the model is 162 square inches. To find the actual area, multiply the model's area by the area scale factor:

\[
\text{Actual Area} = 162 \, \text{square inches} \times 484 = 78408 \, \text{square feet}
\]

### Step 4: Round to the nearest square foot
The calculated area is 78,408 square feet. Rounding to the nearest square foot:

\[
\text{Actual Area} = 78,408 \, \text{square feet}
\]

Thus, the area of the actual roof is approximately **78,408 square feet**.

Would you like more details or have any questions?

Here are some related questions to explore:
1. What is the volume of a structure if the model's volume is given?
2. How does the scale affect the perimeter of the roof?
3. What would be the scale factor for a model with a different scale, say 1 inch:10 feet?
4. How do scale models help in architectural design and planning?
5. How would the dimensions change if the scale were doubled?

**Tip**: For two-dimensional models, always remember that area scales by the square of the linear scale factor!

Learn how to calculate the actual area of a roof from a scale model with a ratio of 1 inch:22 feet. The model roof measures 162 square inches, and this step-by-step solution shows how to use the scale factor to find the actual area, rounding to the nearest square foot. Ideal for high school students studying proportions and scale models.

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