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Difficulty: EasyArea of Two-Dimensional Shapes

A circular rug has an area of 16π16\pi square feet. If the radius of the rug is tripled, what is the area, in square feet, of the new rug?

  1. A
    32π32\pi
  2. B
    48π48\pi
  3. 144π144\piAnswer
  4. D
    64π64\pi

Answer

The area of the new rug is 144π144\pi square feet.
The area of a circle scales with the square of its radius. If the radius is tripled, the area is multiplied by 32=93^2 = 9. Since the original area is 16π16\pi square feet, the area of the new rug is 16π×9=144π16\pi \times 9 = 144\pi square feet.

Step-by-Step Solution

1
Determine the relationship between the linear scale factor and the area scale factor of a circle.
If the radius of a circle is multiplied by a scale factor kk, the area of the circle is multiplied by k2k^2.
The area of a circle is proportional to the square of its radius (A=πr2A = \pi r^2).
2
Find the area scale factor when the radius is tripled.
Since the radius is multiplied by 33, the area scale factor is 32=93^2 = 9.
Squaring the linear scale factor of 3 gives the quadratic scale factor for the area.
3
Calculate the area of the new rug.
The new area is 16π×9=144π16\pi \times 9 = 144\pi square feet.
Multiplying the original area of 16π16\pi square feet by the area scale factor of 9 yields the final scaled area.

Key Concept

Area scaling of two-dimensional shapes
Estimated Time:1m 0s
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Topics

Area of Two-Dimensional Shapes

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