hazelnut01 wrote:A rectangular wooden dowel measures 4 inches by 1 inch by 1 inch. If the dowel is painted on all surfaces and then cut into 1/2-inch cubes, what fraction of the resulting cube faces are painted?
A. 1/3
B. 3/8
C. 7/16
D. 1/2
E. 9/16
Source : Manhattan Prep
OA=B
Nice question.
We can answer it by calculating total surfaces areas.
BEFORE the cuts
There are 6 faces to the dowel, and each face is a rectangle.
4 of the faces have dimensions 4 by 1. So, the area of each rectangle = (4)(1) = 4
So, the total area of those 4 rectangles = (4)(4) = 16
The remaining 2 faces have dimensions 1 by 1. So, the area of each rectangle (square) = (1)(1) = 1
So, the total area of those 2 rectangles = (2)(1) = 2
So, BEFORE the cuts, the TOTAL surface area of the dowel = 16 + 2 =
18
In other words,
there are 18 square inches of paint.
AFTER the cuts
Each cube has dimensions 1/2 by 1/2 by 1/2
So, the VOLUME of each cube = (1/2)(1/2)(1/2) = 1/8
BEFORE the cut, the VOLUME of the dowel = (4)(1)(1) = 4
So, the NUMBER of cubes = 4/(1/8) = 32
So, after the cuts, there are
32 mini cubes
Each individual mini-cube has 6 sides, and each side is a 1/2 by 1/2 square.
So, the area of ONE square = (1/2)(1/2) = 1/4
So, the total surface area of one mini cube = (6)(1/4) = 6/4 = 3/2
So, the TOTAL surface are of all
32 mini cubes = (
32)(3/2) =
48 square inches
So, the 32 mini cubes have a TOTAL surface are of
48 square inches, and
18 square inches are painted.
So, the correct answer is
18/
48
= 3/8
=
B
Cheers,
Brent
