SOLUTION: A rectangular mosaic is to be constructed from 70 congruent square tiles, each of side length 6 inches. Determine the minimum perimeter of the mosaic

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Question 1153234: A rectangular mosaic is to be constructed from 70 congruent square tiles, each of side length 6 inches. Determine the minimum perimeter of the mosaic
Found 2 solutions by josmiceli, greenestamps:
Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
Let the number of tiles on each side = +x+ and +y+
You are given that:
+x%2Ay+=+70+
You need to minimize the perimeter:
+P+=+2x+%2B+2y+
and +y+=+70%2Fx+, so
+P+=+2x+%2B+2%2A%28+70%2Fx+%29+
+P+=+2x+%2B+140%2Fx+
+x+ has to be a whole number, and +P+ has to
be a whole number, so +140%2Fx+ is a whole number.
(1) +x+=+1+
(2) +x+=+2+
(3) +x+=+7+
(4) +x+=+10+
(5) +x+=+20+
(6) +x+=+70+
(7) +x+=+140+
are the possible choices
The perimeters are:
+P%5B1%5D+=+142+
+P%5B2%5D+=+74+
+P%5B3%5D+=+34+
+P%5B4%5D+=+34+
+P%5B5%5D+=+142+
+P%5B6%5D+=+281+
---------------------
So, +x+=+7+ or +x+=+10+ gives
the minimum perimeter
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In inches, this is:
+P%5Bmin%5D+=+2x+%2B+140%2Fx+
+P%5Bmin%5D+=%28++2%2A7+%2B+140%2F7+%29%2A6+
+P%5Bmin%5D+=+34%2A6+
+P%5Bmin%5D+=+204+ in.
----------------------------
also:
+P%5Bmin%5D+=+%28+2%2A10+%2B+140%2F10+%29%2A6+
+P%5Bmin%5D+=+204+ in.
---------------------------
Definitely get a 2nd opinion if needed

Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


The solution from the other tutor is fine; but you can get to the answer without nearly as much work as she shows using some basic concepts.

Let the dimensions of the rectangle be x and y. Then we have

xy+=+70

with x and y whole numbers; and we are to minimize the perimeter, 2x+2y, which is the same as minimizing x+y.

A general principle for minimizing x+y when the product xy is fixed is that the difference between x and y should be minimum.

In this problem, with a product of 70 and both numbers being whole numbers, that means the dimensions are 10 and 7.

Then the perimeter (in number of blocks) is 2(10+7) = 34; and the perimeter in inches is 34*6 = 204.