Cracking the SAT Subject Test in Math 2, 2nd Edition (B = area of base) Connect the Dots Notice that the volume of a pyramid is just one third of the volume of a prism Make memorizing easy! It’s not r[.]
(B = area of base) Connect the Dots Notice that the volume of a pyramid is just onethird of the volume of a prism Make memorizing easy! It’s not really possible to give a general formula for the surface area of a pyramid because there are so many different kinds At any rate, the information is not generally tested on the SAT Subject Test in Math 2 If you should be called upon to figure out the surface area of a pyramid, just figure out the area of each face using polygon rules, and add them up TRICKS OF THE TRADE Here are some common ways ETS likes to use solid geometry on the SAT Subject Test in Math As you will see, many of these questions test concepts you know (including plane geometry!) in unfamiliar ways Triangles in Rectangular Solids Many questions about rectangular solids are actually testing triangle rules Such questions generally ask for the lengths of the diagonals of a box’s faces, the long diagonal of a box, or other lengths These questions are usually solved using the Pythagorean Theorem and the Super Pythagorean Theorem that finds a box’s long diagonal (see the section on Rectangular Solids) DRILL 1: TRIANGLES IN RECTANGULAR SOLIDS Here are some practice questions using triangle rules in rectangular solids The answers can be found in Part IV 17 What is the length of the longest line that can be drawn in a cube of volume 27 ? (A) 3.0 (B) 4.2 (C) 4.9 (D) 5.2 (E) 9.0 22 In the rectangular solid shown, if AB = 4, BC = 3, and BF = 12, what is the perimeter of triangle EDB ? (A) 27.33 (B) 28.40 (C) 29.20 (D) 29.50 (E) 30.37 25 In the cube above, M is the midpoint of BC, and N is the midpoint of GH If the cube has a volume of 1, what is the length of MN ? (A) 1.23 (B) 1.36 (C) 1.41 (D) 1.73 (E) 1.89 41 In the cube shown above with side length s, what is the area of the plane inside the cube defined by D, E, and G ? (A) s2 (B) (C) (D) s2 (E) Volume Questions Many solid geometry questions test your understanding of the relationship between a solid’s volume and its other dimensions— sometimes including the solid’s surface area To solve these questions, just plug the numbers you’re given into the solid’s volume formula DRILL 2: VOLUME Try the following practice questions The answers can be found in Part IV The volume and surface area of a cube are equal What is the length of an edge of this cube? (A) (B) (C) (D) (E) A cube has a surface area of 6x What is the volume of the cube? (A) (B) (C) 6x2 (D) 36x2 (E) x3 10 A rectangular solid has a volume of 30, and its edges have integer lengths What is the greatest possible surface area of this solid? (A) 62 (B) 82 (C) 86 (D) 94 (E) 122 15 The water in Allegra’s swimming pool has a depth of 7 feet If the area of the pentagonal base of the pool is 150 square feet, then what is the volume, in cubic feet, of the water in her pool? (A) 57 (B) 50 (C) 1,050 (D) 5,250 (E) It cannot be determined from the information given 26 A sphere has a radius of r If this radius is increased by b, then the surface area of the sphere is increased by what amount? (A) b2 (B) 4πb2 (C) 8πrb + 4πb2 (D) 8πrb + 2rb + b2 (E) 4πr2b2 35 If the pyramid shown has a square base with edges of length b, and b = 2h, then which of the following is the volume of the pyramid? (A) (B) (C) 4h3 (D) 8h2 − h (E) 42 A sphere of radius 1 is totally submerged in a cylindrical tank of radius 4, as shown The water level in the tank rises a distance of h What is the value of h ? (A) 0.072 (B) 0.083 (C) 0.096 (D) 0.108 (E) 0.123 Inscribed Solids Some questions on the SAT Subject Test in Math will be based on spheres inscribed in cubes or cubes inscribed in spheres (these are the most popular inscribed shapes) Occasionally you may also see a rectangular solid inscribed in a sphere, or a cylinder inscribed in a rectangular box, etc The trick to these questions is always figuring out how to get from the dimensions of one of the solids to the dimensions of the other Following are a few basic tips that can speed up your work on inscribed solids questions • When a cube or rectangular solid is inscribed in a sphere, the long diagonal of that solid is equal to the diameter of the sphere • When a cylinder is inscribed in a sphere, the sphere’s diameter is equal to the diagonal of the rectangle formed by the cylinder’s heights and diameter • When a sphere is inscribed in a cube, the diameter of the sphere is equal to the length of the cube’s edge • If a sphere is inscribed in a cylinder, both solids have the same diameter Most inscribed solids questions fall into one of the preceding categories If you run into a situation not covered by these tips, just look for the way to get from the dimensions of the inner shape to those of the external shape, or vice versa DRILL 3: INSCRIBED SOLIDS Here are some practice inscribed solids questions The answers can be found in Part IV 17 A rectangular solid is inscribed in a sphere as shown If the dimensions of the solid are 3, 4, and 6, then what is the radius of the sphere? (A) 2.49 (B) 3.91 (C) 4.16 (D) 5.62 (E) 7.81 19 A cylinder is inscribed in a cube with an edge of length 2 What volume of space is enclosed by the cube but not by the cylinder? (A) 1.41 (B) 1.56 (C) 1.72 (D) 3.81 (E) 4.86 25 A cone is inscribed in a cube of volume 1 in such a way that its base is inscribed in one face of the cube What is the volume of the cone? (A) 0.21 (B) 0.26 (C) 0.33 (D) 0.42 (E) 0.67 37 Cube A has a volume of 1,000 Sphere S is inscribed inside Cube A Cube B (not shown) is inscribed inside Sphere S What is the surface area of Cube B ? (A) 5.774 (B) 33.333 (C) 192.450 (D) 200 (E) 600 Solids Produced by Rotation Three types of solids can be produced by the rotation of simple twodimensional shapes—spheres, cylinders, and cones Questions about solids produced by rotation are generally fairly simple; they usually test your ability to visualize the solid generated by the rotation of a flat shape Sometimes, rotated solids questions begin with a shape in the coordinate plane—that is, rotated around one of the axes or some other line Practice will help you figure out the dimensions of the solid from the dimensions of the original flat shape A sphere is produced when a circle is rotated around its diameter This is an easy situation to work with, as the sphere and the original circle will have the same radius Find the radius of the circle, and you can figure out anything you want to about the sphere A cylinder is formed by the rotation of a rectangle around a central line or one edge A cone is formed by rotating a right triangle around one of its legs (think of it as spinning the triangle) or by rotating an isosceles triangle around its axis of symmetry Another way of thinking about it is if you spun the triangle in the first figure above around the y-axis (so you’re rotating around the leg that’s sitting on the y-axis) you would get the second figure Likewise, if you spun the third figure above around the x-axis (so you’re rotating around the axis of symmetry), you would end up with the fourth figure DRILL 4: ROTATIONAL SOLIDS Try these rotated solids questions for practice The answers can be found in Part IV 19 What is the volume of the solid generated by rotating rectangle ABCD around AD ? (A) 15.7 (B) 31.4 (C) 62.8 (D) 72.0 (E) 80.0 29 If the triangle created by OAB is rotated around the xaxis, what is the volume of the generated solid? (A) 15.70 (B) 33.33 (C) 40.00 (D) 47.12 (E) 78.54 40 What is the volume generated by rotating square ABCD around the y-axis? (A) 24.84 (B) 28.27 (C) 42.66 (D) 56.55 (E) 84.82 Changing Dimensions Some solid geometry questions will ask you to figure out what happens to the volume of a solid if all of its lengths are increased by a certain factor or if its area doubles, and so on To answer questions of this type, just remember a basic rule When the lengths of a solid are increased by a certain factor, the surface area of the solid increases by the square of that factor, and the volume increases by the cube of that factor This rule is true only when the solid’s shape doesn’t change—its length must increase in every dimension, not just one For that reason, cubes and spheres are most often used for this type of question because their shapes are constant In the illustration on the previous page, a length is doubled, which means that the corresponding area is 4 times as great, and the volume is 8 times as great If the length had been tripled, the area would have increased by a factor of 9, and the volume by a factor of 27 DRILL 5: CHANGING DIMENSIONS Try these practice questions The answers can be found in Part IV If the radius of sphere A is one-third as long as the radius of sphere B, then the volume of sphere A is what fraction of the volume of sphere B ? (A) (B) (C) (D) (E) A rectangular solid with length l, width w, and height h has a volume of 24 What is the volume of a rectangular solid with length , width , and height ? (A) 18 (B) 12 (C) (D) (E) 10 If the surface area of a cube is increased by a factor of 2.25, then its volume is increased by what factor? (A) 1.72 (B) 3.38 (C) 4.50 (D) 5.06 (E) 5.64 33 The volume of the cylinder shown above is x If the radius of the cylinder is doubled and the height is halved, then which of the following is the volume of the new cylinder? (A) (B) x (C) 2x (D) x2 (E) 46 In the regular hexagonal prism shown above, if the lengths and widths of the rectangular faces were doubled and the hexagonal faces changed accordingly, then the resulting volume would be how many times the original volume? (A) 64 (B) 32 (C) (D) (E) ... Inscribed Solids Some questions on the SAT Subject Test in Math will be based on spheres inscribed in cubes or cubes inscribed in spheres (these are the most popular inscribed shapes) Occasionally... (E) 5.64 33 The volume of the cylinder shown above is x If the radius of the cylinder is doubled and the height is halved, then which of the following is the volume of the new cylinder? (A) (B)... its axis of symmetry Another way of thinking about it is if you spun the triangle in the first figure above around the y-axis (so you’re rotating around the leg that’s sitting on the y-axis) you would get the second