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Zorluk: ZorScale Drawings

A city planner is reviewing a map of a proposed municipal park. The map uses a scale where 12\frac{1}{2} inch represents 3030 feet. If the park occupies an area of 1515 square inches on the map, what will be the actual area of the park in square yards?

  1. A
    300
  2. 6,000Cevap
  3. C
    18,000
  4. D
    54,000

Cevap

The actual area of the park is 6,000 square yards.
To solve this problem efficiently, first convert the linear scale to yards: 12 inch=30 feet\frac{1}{2}\text{ inch} = 30\text{ feet} means 1 inch=60 feet1\text{ inch} = 60\text{ feet}, which equals 20 yards20\text{ yards}. Because we are dealing with area, we must square the linear scale to find the area scale factor: 1 square inch=202 square yards=400 square yards1\text{ square inch} = 20^2\text{ square yards} = 400\text{ square yards}. Finally, multiply the given map area by this factor: 15 square inches×400=6,000 square yards15\text{ square inches} \times 400 = 6,000\text{ square yards}.

Adım Adım Çözüm

1
Determine the linear scale factor for one full inch.
1 inch=60 feet1\text{ inch} = 60\text{ feet}
The map scale is given as 12 inch=30 feet\frac{1}{2}\text{ inch} = 30\text{ feet}. Multiplying both sides by 2 gives the equivalent measurement for 1 inch.
2
Convert the linear scale into yards to match the requested final unit.
1 inch=20 yards1\text{ inch} = 20\text{ yards}
Since there are 3 feet in a yard, dividing 60 feet by 3 yields 20 yards. This simplifies calculations by converting units early.
3
Calculate the area scale factor by squaring the linear scale.
1 square inch=400 square yards1\text{ square inch} = 400\text{ square yards}
Area is a two-dimensional measurement. The linear scale must be squared (20×20=40020 \times 20 = 400) to establish the relationship between map area and actual area.
4
Calculate the actual area of the park.
15×400=6,000 square yards15 \times 400 = 6,000\text{ square yards}
Multiply the map area by the area scale factor to find the true dimensions.

Anahtar Kavram

Scale Factors for Area and Unit Conversions

Alternatif Yöntem

Instead of converting to yards early, you can find the actual area in square feet first. The area scale factor in feet is 602=3,600 square feet per square inch60^2 = 3,600\text{ square feet per square inch}. Multiply by the map area: 15×3,600=54,000 square feet15 \times 3,600 = 54,000\text{ square feet}. Then convert to square yards by dividing by 9: 54,000÷9=6,000 square yards54,000 \div 9 = 6,000\text{ square yards}.
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