Jump to a New ChapterIntroduction to the SAT IIContent and Format of the SAT II Math IICStrategies for SAT II Math IICMath IIC 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 surface area of a rectangular solid is:

where l, w, and h are the dimensions of the solid. To answer the question, substitute 2l, 2w, and 2h for l, w, and h:

When the measurements of the rectangular solid are doubled, the surface area of the solid is multiplied by 4.

2.      D

The formula for the volume of a cylinder is V = πr2h, where r is the radius of its circular base and h is its height. The height of the cylinder is given, so we just need to find out its radius. The best way to do this is to sketch a diagram. If you look at the figure below, you can see that the diameter of the sphere, d, is the diagonal of the cylinder. The diagonal of the cylinder and its height, together with the diameter of the cylinder’s circular base as the third side, make up a right triangle:

We could use the Pythagorean theorem to calculate the diameter of the cylinder base, but you should recognize that the triangle formed is a multiple of a 3-4-5 triangle, a Pythagorean triple. Therefore, since h = 2 4 = 8 and d = 2r = 2 5 = 10, the diameter of the cylinder base is 2 3 = 6. This means that the radius of the base is 3, and the volume of the cylinder can now be calculated:

3.      C

The formula for the volume of a cube is Vc = s3, where s is the length of a side of the cube. The formula for the volume of a sphere is Vs = 4πr3⁄3, where r is the radius of the sphere. Therefore, since we know that Vc = 3Vs, we can substitute in the formulas for the two solids and solve for s/r :

4.      C

When the right triangle is rotated around the leg of length 12, a cone is formed with radius 5 and height 12. The formula for the surface area of a cone is πr2 + πrl, where r is the radius and l is the lateral length of the cone. So, in order to calculate the surface area, we need to first find l:

As you can see, l is just the hypotenuse of the right triangle, which is a 5-12-13 triangle (a Pythagorean triple). So l = 13, and we can plug in values to find the surface area of the cone:

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