Jump to a New ChapterIntroduction to the SAT IIIntroduction to SAT II Math ICStrategies for SAT II Math ICMath IC FundamentalsAlgebraPlane GeometrySolid GeometryCoordinate GeometryTrigonometryFunctionsStatisticsMiscellaneous MathPractice Tests Are Your Best Friends
 7.1 Prisms 7.2 Solids That Aren’t Prisms 7.3 Relating Length, Surface Area, and Volume 7.4 Inscribed Solids

 7.5 Solids Produced by Rotating Polygons 7.6 Key Formulas 7.7 Review Questions 7.8 Explanations
Explanations

1.      C

The formula for the length of the diagonal of a rectangular solid is where l, w, and h are the dimensions of the solid. Substitute 2l, 2w, and 2h for these values:

The length of the diagonal doubles, just like the rectangular solid’s other dimensions.

2.      D

The volume of a cylinder is given by the formula . In the case of this cylinder, the radius is equal to height (r = h), so

Since we have a value only for the surface area of this cylinder, to find r, we must use the surface area formula, which is Because r = h in this cylinder, the surface area of this cylinder is

With the information given by the question, we can find the radius by setting either r2 = 100π or r2 = 100π⁄4π = 25. Either way, the radius of the cylinder works out to 5, which means that the volume of the cylinder is π53 = 125π.

3.      C

The formula for a cone’s surface area is πr2 + πrl. A cone’s volume is 1/ 3πr2h. So if the dimensions of a cone are multiplied by the same factor, a,

then its surface area multiplies by the square of that factor. If the dimensions of a cone are multiplied by the same factor, the volume becomes multiplied by the cube of that factor:

In general, for solids, if each dimension of a cone is multiplied by the same factor, the solid’s surface area is multiplied by the square of that factor, and its volume increases by the cube of that factor. If the surface area of Cone A doubles, its dimensions are multiplied by a factor of Thus, the cone’s volume is multiplied by a factor of Cone A’s new volume is

4.      E

If the rectangle is rotated about a side of length 4, then the height of the cylinder will be 4 and the radius will be 6.

Once you visualize the cylinder, you can plug in the values for the volume of a cylinder:

 Jump to a New ChapterIntroduction to the SAT IIIntroduction to SAT II Math ICStrategies for SAT II Math ICMath IC FundamentalsAlgebraPlane GeometrySolid GeometryCoordinate GeometryTrigonometryFunctionsStatisticsMiscellaneous MathPractice Tests Are Your Best Friends
Test Prep Centers
SparkCollege
 College Admissions Financial Aid College Life