新托福考试辅导_ Relating Length, Surface Area, and Volume

Example 2


	If a sphere’s radius is halved, by what factor does its volume decrease?

The radius of the sphere is multiplied by a factor of ¹⁄₂ (or divided by a factor of 2), and so its volume multiplies by the cube of that factor: (¹⁄₂)³ = ¹⁄₈. Therefore, the volume of the sphere is multiplied by a factor of ¹⁄₈ (divided by 8), which is the same thing as decreasing by a factor of 8.

Example 3


	A rectangular solid has dimensions x y z (these are its length, width, and height), and a volume of 64. What is the volume of a rectangular solid of dimensions ^x /₂ ^y /₂ z?

If this rectangular solid had dimensions that were all one-half as large as the dimensions of the solid whose volume is 64, then its volume would be (¹⁄₂)³

64 = ¹⁄₈

64 = 8. But dimension z is not multiplied by ¹⁄₂ like x and y. To answer a question like this one, you should use the volume formula for rectangular solids: Volume = l

h. It is given in the question that xyz = 64. So, ^x⁄₂

^y⁄₂

z = ¹⁄₄

xyz = ¹⁄₄

64 = 16.

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