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Zorluk: OrtaArea of Two-Dimensional Shapes

A property line is in the shape of a trapezoid with parallel sides of length 2020 meters and 3030 meters, and a height of 1212 meters. A second property is geometrically similar to the first, where each linear dimension of the second property is 1.51.5 times the corresponding dimension of the first property. What is the area, in square meters, of the second property?

  1. A
    450
  2. 675Cevap
  3. C
    900
  4. D
    1,350

Cevap

675
The area of the first property is calculated using the trapezoid area formula: A=12(b1+b2)hA = \frac{1}{2}(b_1 + b_2)h. Substituting the given values gives A=12(20+30)(12)=300A = \frac{1}{2}(20 + 30)(12) = 300 square meters. Since the second property is similar to the first with a linear scale factor of 1.51.5, its area is scaled by 1.52=2.251.5^2 = 2.25. Therefore, the area of the second property is 300×2.25=675300 \times 2.25 = 675 square meters.

Adım Adım Çözüm

1
Calculate the area of the first trapezoidal property.
300300 square meters
Using the trapezoid area formula A=12(b1+b2)hA = \frac{1}{2}(b_1 + b_2)h, we find A=12(20+30)(12)=300A = \frac{1}{2}(20 + 30)(12) = 300.
2
Determine the area scale factor for the similar property.
2.252.25
Since the second property is similar to the first with a linear scale factor of 1.51.5, its area scales by the square of this factor: (1.5)2=2.25(1.5)^2 = 2.25.
3
Calculate the area of the second property.
675675 square meters
Multiplying the original area by the area scale factor yields 300×2.25=675300 \times 2.25 = 675.

Anahtar Kavram

Area of similar two-dimensional shapes scales by the square of the linear scale factor.
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Konular

Area of Two-Dimensional Shapes

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