Three-Dimensional Geometry - Geometry and Measurement - REVIEW OF MAJOR TOPICS - Barron's SAT Subject Test Math Level 2

Barron's SAT Subject Test Math Level 2, 10th Edition (2012)

Part 2. REVIEW OF MAJOR TOPICS

Chapter 2. Geometry and Measurement

2.2 Three-Dimensional Geometry

SURFACE AREA AND VOLUME

The Level 2 test may have problems involving five types of solids: prisms, pyramids, cylinders, cones, and spheres. Rectangular solids and cubes are special types of prisms. In a rectangular solid, the bases are rectangles, so any pair of opposite sides can be bases. In a cube, the bases are squares, and the altitude is equal in length to the sides of the base.

On the Level 2 test, all figures are assumed to be right solids. Formulas for the volumes of regular pyramids, circular cones, and spheres and formulas for the lateral areas of circular cones and spheres are given in the test book. For convenience, formulas for the volumes and lateral areas of all of these figures are summarized below.

Solid

Volume

Surface Area

Prism

V = Bh

SA = Ph + 2B

Rectangular solid  

V = lwh

SA = 2lw + 2lh + 2wh

Cube

V = s3

SA = 6s2

Pyramid

r10

r11

Cylinder

V = πr2h

SA = 2πrh + 2πr2

Cone

r12  

SA = πrL + πr2

Sphere

r13

SA = 4πr2

The variables in these formulas are defined as follows:

V = volume

h = altitude

SA = surface area

l = length

B = base area

w = width

P = base perimeter  

r = radius

L = slant height

s = side length

Using the notation above, the formula for the length of a diagonal of a rectangular solid is r14, which is derived from two applications of the Pythagorean theorem.

EXAMPLES

1. The length, width, and height of a right prism are 9, 4, and 2, respectively. What is the length of the longest segment whose endpoints are vertices of the prism?

The longest such segment is a diagonal of the prism. The length of this diagonal is

r15

One type of problem that might be found on a Math Level 2 test is to find the volume of the space between two solids, one inscribed in the other. This is solved by finding the difference of volumes. In this type of problem, you are given only some of the necessary dimensions. You must use these dimensions to find the others.

2. A sphere with diameter 10 meters is inscribed in a cube. What is the volume of the space between the sphere and the cube?

You need to subtract the volume of the sphere from the volume of the cube. The radius of the sphere is 5 and the volume is r16. Since the sphere is inscribed in the cube, the side length of the cube is 10 m and its volume is 1000 m3. The volume of the space between them is 476.4 m3.

You might be asked to find a dimension of a figure if its volume and surface area are equal.

3. A cylinder has a volume of 500 in3. What is the radius of this cylinder if its altitude equals the diameter of its base?

The formula for the volume of a cylinder is V = πr2h . If h = 2r, this formula becomes V = 2πr3. Substitute 500 for V and solve for r to get r = 4.3 in.

A third type of problem involves percent changes in the dimensions of a figure.

4. If the volume of a right circular cone is reduced by 15% by reducing its height by 5%, by what percent must the radius of the base be reduced?

If the volume of the cone is reduced by 15%, then 85% of its original volume remains. If its height is reduced by 5%, 95% of its original height remains. Use the formula for the volume of a cone, and let p be the proportion of the radius that remains,

r17

This equation simplifies to 0.85 = 0.95p2, so p r8 0.946. Therefore, the radius must be reduced by 5.4%.

A solid figure can be obtained by rotating a plane figure about some line in the plane that does not intersect the figure.

EXERCISES

1. The figure below shows a right circular cylinder inscribed in a cube with edge of length x . What is the ratio of the volume of the cylinder to the volume of the cube?

SAT_SUBJECT_TEST_MATH_LEVEL_2_9TH_ED_0138_001

 (A) r18

 (B) r20

 (C) r21

 (D) r22

 (E) r23

2. The volume of a right circular cylinder is the same numerical value as its total surface area. Find the smallest integral value for the radius of the cylinder.

 (A) 1

 (B) 2

 (C) 3

 (D) 4

 (E) This value cannot be determined.

3. The length, width, and height of a rectangular solid are 5 cm, 3 cm, and 7 cm, respectively. What is the length of the longest segment whose endpoints are vertices of the rectangular solid?

 (A) 5.8 cm

 (B) 7.6 cm

 (C) 8.6 cm

 (D) 9.1 cm

 (E) 15 cm

COORDINATES IN THREE DIMENSIONS

The coordinate plane can be extended by adding a third axis, the z-axis, which is perpendicular to the other two. Picture the corner of a room. The corner itself is the origin. The edges between the walls and the floor are the x-and y-axes. The edge between the two walls is the z-axis. The first octant of this three-dimensional coordinate system and the point (1,2,3) are illustrated below.

SAT_SUBJECT_TEST_MATH_LEVEL_2_9TH_ED_0139_001

A point that has zero as any coordinate must lie on the plane formed by the other two axes. If two coordinates of a point are zero, then the point lies on the nonzero axis. The three-dimensional Pythagorean theorem yields a formula for the distance between two points (x1,y1,z1) and (x2,y2,z2) in space:

SAT_SUBJECT_TEST_MATH_LEVEL_2_9TH_ED_0139_002

A three-dimensional coordinate system can be used to graph the variable z as a function of the two variables x and y, but such graphs are beyond the scope of the Level 2 Test.

EXAMPLES

1. The distance between two points in space A (2, y, –3) and B (1, –1, 4) is 9. Find the possible values of y .

Use the formula for the distance between two points and set this equal to 9:

SAT_SUBJECT_TEST_MATH_LEVEL_2_9TH_ED_0139_003

Square both sides and simplify to get (y + 1)2 = 31. Therefore, y r8 4.6 or y r8 –6.6.

A sphere is the set of points in space that are equidistant from a given point. If the given point is (a, b, c) and the given distance is r, an equation of the sphere is (xa)2 + (yb)2 + (zc)2 = r2.

2. Describe the graph of the set of points (x, y, z) where
(x – 6)2 + (y + 3)2 + (z – 2)2 = 36.

This equation describes the set of points whose distance from (6, –3, 2) is 6. This is a sphere of radius 6 with center at (6, –3, 2).

If only two of the variables appear in an equation, the equation describes a planar figure. The third variable spans the entire number line. The resulting three-dimensional figure is a solid that extends indefinitely in both directions parallel to the axis of the variable that is not in the equation, with cross sections congruent to the planar figure.

3. Describe the graph of the set of points (x, y, z) where x2 + z2 = 1.

Since y is not in the equation, it can take any value. When restricted to the xz plane, the equation is that of a circle with radius 1 and center at the origin. Therefore in xyz space, the equation represents a cylindrical shape, centered at the origin, that extends indefinitely in both directions along the y-axis.

A solid figure can also be obtained by rotating a plane figure about some line in the plane that does not intersect the figure.

4. If the segment of the line y = –2x + 2 that lies in quadrant I is rotated about the y-axis, a cone is formed. What is the volume of the cone?

As shown in the figure below on the left, the part of the segment that lies in the first quadrant and the axes form a triangle with vertices at (0,0), (1,0), and (0,2). Rotating this triangle about the y-axis generates the cone shown in the figure below on the right.

SAT_SUBJECT_TEST_MATH_LEVEL_2_9TH_ED_0140_001

The radius of the base is 1, and the height is 2. Therefore, the volume is

SAT_SUBJECT_TEST_MATH_LEVEL_2_9TH_ED_0140_002

EXERCISES

1. The distance between two points in space, P (x,–1,–1) and Q (3,–3,1), is 3. Find the possible values of x .

 (A) 1 or 2

 (B) 2 or 3

 (C) –2 or –3

 (D) 2 or 4

 (E) –2 or –4

2. The point (–4,0,7) lies on the

 (A) y-axis

 (B) xy plane

 (C) yz plane

 (D) xz plane

 (E) z -axis

3. The region in the first quadrant bounded by the line 3x + 2y = 7 and the coordinate axes is rotated about the x-axis. What is the volume of the resulting solid?

 (A) 8 units3

 (B) 20 units3

 (C) 30 units3

 (D) 90 units3

 (E) 120 units3

Answers and Explanations

Surface Area and Volume

1. (C) The volume of the cube is x3. The radius of the cylinder is r24, and its height is x . Substitute these into the formula for the volume of a cylinder:

SAT_SUBJECT_TEST_MATH_LEVEL_2_9TH_ED_0141_001

2. (C) V = πr2h, and total area = 2πr2 + 2πrh . Setting these two equal yields rh = 2r + 2h . Therefore, r25. Since h must be positive, the smallest integer value of r is 3.

3. * (D) The length of the longest segment is r26

Coordinates in Three Dimensions

1. (D) The square of the distance between P and Q is 9, so

(x – 3)2 + (–1 – (–3))2 + (–1 – 1)2 = 9, or (x – 3)2 = 1.

Therefore, x – 3 = ±1, so x = 2 or 4.

2. (D) Since the y -coordinate is zero, the point must lie in the xz plane.

3. * (C) The line 3x + 2y = 7 has x -intercept r27 and y -intercept r28. The part of this line that lies in the first quadrant forms a triangle with the coordinate axes. Rotating this triangle about the x -axis produces a cone with radius r28 and height r27. The volume of this cone is r29.