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) (Show Source): You can put this solution on YOUR website!
Let the number of tiles on each side = and
You are given that:
You need to minimize the perimeter:
and , so
has to be a whole number, and has to
be a whole number, so is a whole number.
(1)
(2)
(3)
(4)
(5)
(6)
(7)
are the possible choices
The perimeters are:
---------------------
So, or gives
the minimum perimeter
---------------------------
In inches, this is:
in.
----------------------------
also:
in.
---------------------------
Definitely get a 2nd opinion if needed
Answer by greenestamps(13200) (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
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.
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