'manpreet singh wrote:Suppose five circles, each 4 inches in diameter, are cut from a rectangular strip of paper 12 inches long. If the least amount of paper is to be wasted, what is the width of the paper strip?
(A) 5
(B) 4 + 2√3
(C) 8
(D) 4(1 + √3)
(E) not enough information
Let's begin by sketching a scenario that minimizes paper waste:
At this point, we should be able to see that the height of the rectangle is
less than 8 inches. So, we can eliminate C, D and E.
Can we eliminate any other answer choices?
To find out, let's go a little further.
Since the circles have radius 2, we can add 2 measurements to our sketch. So, all we need to do now is find the length of that missing blue line
At this point, we should recognize that the length of the blue line must be at least 2 inches (the radius of a circle), so we can eliminate answer choice A, which leaves us with
B, the correct answer. . . DONE.
For those who are interested in the complete solution (in case the answer choices weren't so nice), here it goes . . .
Notice that, if we connect the centers of 3 circles, we get an equilateral triangle where each side has length 4.
Now focus on the following right triangle.
From here, we can find the length of blue side using the Pythagorean Theorem, or we can use the fact that we have a special 30-60-90 right triangle and apply the ratios for those kinds of triangles.
Either way, we'll conclude that the blue side has length 2√3
When we add this to our sketch . . .
. . . we see that the height of this rectangle = [spoiler]4 + 2√3 = B[/spoiler]
Cheers,
Brent
