SAT Mc Graw Hill 2011

Volume

The SAT math section may include a question

or two about volumes. Remember two things:

• The volume of a container is nothing more
  than the number of “unit cubes” it can hold.


• The only volume formulas you will need are
  given to you in the “Reference Information”
  on every math section.

Example:

How many rectangular bricks measuring 2 inches

by 3 inches by 4 inches must be stacked together

(without mortar or any other material) to create

a solid rectangular box that measures 15 inches by

30 inches by 60 inches?

Don’t be too concerned with howthe bricks could be
stacked to make the box; there are many possible
arrangements, but the arrangement doesn’t affect the
answer. All you need to know is that it can be done. If
so, just looking at the volumes is enough: if you use
nbricks, then the box must have a volume that is n
times larger than the volume of one brick. Each brick
has a volume of 2 × 3 × 4 =24 cubic inches. The box
has a volume of 15 × 30 × 60 =27,000 square inches.
The number of bricks, then, must be 27,000/24 =
1,125.

Lesson 7: Volumes and 3-D Geometry

3-D Distances

If you are trying to find the length of a line seg-

ment in three dimensions, look for a right tri-

angle that has that segment as its hypotenuse.

Example:

The figure at right shows a cube

with edges of length 4. If

point Cis the midpoint of

edgeBD,what is the length

ofAC

–––

?

Draw segment CE

––

to see that AC

––

is the hypotenuse of right tri-

angle∆AEC.

LegAE

––

has a length of 4, and

legEC

––

is the hypotenuse of

right triangle ∆EBC,with legs

of length 2 and 4. Therefore,

so.

One possible shortcut for finding lengths in

three dimensions is the three-dimensional dis-

tance formula:

If you think of point Ain the cube above as being the

origin (0, 0, 0), then point Ccan be considered to be

(4, 4, 2). The distance from AtoC,then, is

dxx yy zz=−() 21 +−()+−()

2

21

2

21

2

AC= () 20 += += = 4 20 16 36 6

2

2

EC=+=+=24 416 20^22

B DC

A

4

B DC

A

4

4

2

E

394 McGRAW-HILL’S SAT

40 40 20 16164 36 6

222

()− +−()+−()=++= =.