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44. The correct answer is (B). The neat thing about a cube is that if you have any one feature, e.g.,
volume, edge, diagonal of a face, diagonal of the cube, surface area, you can calculate every other
feature. It is for this reason that cubes are often the focus of test problems. Given that the edge has a
length of 2, we can use the Pythagorean Theorem to find the length of the diagonal of a face:
Now we can find the length of the diagonal of the cube:

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Lesson 8
222
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ARCO
■ SAT II Subject Tests
That is the length of the entire diagonal of the cube. The point that is the center of the cube is the
midpoint of the diagonal of the cube and is 
from each vertex.
45. The correct answer is (D). The trick to this problem is to recognize that:
x
2
+ ax + bx + ab = (x + a)(x + b)
Since (x + a)(x + b) = 0, one or the other factor must be 0. Since x + b = 2, x + a = 0.
46. The correct answer is (E). The formula for calculating the volume of a cylinder is: V = 
πr
2
h
So the volume of the larger cylinder is just that. Now we redefine the dimensions of the smaller
cylinder in terms of r and h:
r = kr
′ so r′ = r/k
h = kh
′ so h′ = h/k
And the volume of the smaller cylinder is:
And the ratio is:
As an alternative, you could assume some numbers. Let the radius and height of the larger cylinder be
4 and 4, respectively, and those of the smaller cylinder 2 and 2. Since r = kr
′ and h = kh′, k must be 2.
Now the larger cylinder has a volume of:
V = 
π (4)
2
4 = 64 
π
And the smaller cylinder a volume of:
V = 
π (2)
2
2 = 8
π
And the ratio of 64
π to 8π is 8 to 1 or simply 8. Now, using k = 2, find an answer choice that has the
value of 8:
(A)
Wrong.
(B)
π
Wrong.
(C)
8
π
Wrong.
(D)
Wrong.
(E)
8
Correct.

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