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Difficulty: MediumRatios, Rates, and Proportions

An architectural design firm is constructing a scale model of a commercial building. On the model, a rectangular conference room measures 6 inches6\text{ inches} in length and 4 inches4\text{ inches} in width. If the actual length of the conference room is 45 feet45\text{ feet}, what is the actual area, in square feet, of the conference room?

  1. A
    180180
  2. B
    675675
  3. C
    1,0801,080
  4. 1,3501,350Answer
  5. E
    2,7002,700

Answer

The actual area of the conference room is 1,3501,350 square feet.
To find the actual area, first set up a proportion using the ratio of model length to actual length (6 inches:45 feet6\text{ inches} : 45\text{ feet}). Solving 645=4w\frac{6}{45} = \frac{4}{w} gives 6w=1806w = 180, so the actual width is w=30 feetw = 30\text{ feet}. Multiplying the actual length (45 feet45\text{ feet}) by the actual width (30 feet30\text{ feet}) gives the total area of 1,350 square feet1,350\text{ square feet}.

Step-by-Step Solution

1
Determine the linear scale factor between the model and the actual building.
The ratio of length is 6 inches:45 feet6\text{ inches} : 45\text{ feet}, which simplifies to 1 inch=7.5 feet1\text{ inch} = 7.5\text{ feet}.
Establishing the linear scale factor allows converting model measurements to actual dimensions.
2
Calculate the actual width of the conference room using a proportion.
\frac{6\text{ in}}{45\text{ ft}} = \frac{4\text{ in}}{w\text{ ft}} \implies 6w = 180 \implies w = 30\text{ feet}.
Corresponding linear dimensions in scale models are directly proportional.
3
Calculate the actual area of the rectangular room.
\text{Area} = 45\text{ ft} \times 30\text{ ft} = 1,350\text{ sq ft}.
The area of a rectangle is found by multiplying its length by its width.

Key Concept

Solving scale model problems by establishing linear proportions before calculating area.
Estimated Time:1m 15s
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Topics

Ratios, Rates, and Proportions

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