A square, where s is an odd integer, is divided into unit squares. All the unit squares along the edges and the 2 diagonals of a square are discarded. Find a fully simplified expression, in terms of s, for the number of unit squares remaining answer.
How many unit squares are needed to cover the figures? Use the unit squares shown.
Figure A shows 2 squares. The sides of the larger square are approximately twice as long as the sides of the smaller square. The smaller square is labeled 1 square unit. Figure B shows 1 squares. The sides of the larger square are approximately 4 times as long as the sides of the smaller square. The smaller square is labeled 1 square unit.
A.
Figure A: 2 square units; Figure B: 4 square units
B.
Figure A: 2 square units; Figure B: 8 square units
C.
Figure A: 4 square units; Figure B: 8 square units
D.
Figure A: 4 square units; Figure B: 16 square units
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2 of 3 Answered
there are 4(s-1) edge squares
there are an additional 2(s-2)-1 diagonal squares. Subtract that from the total area, and you have
s^2 - (4(s-1) + 2(s-2)-1)
You can simplify that...
To solve this problem, let's break it down step by step.
First, we need to determine the dimensions of the square in terms of "s". Since the question states that "s" is an odd integer, we know that the square has an odd side length. Let's say the side length is "s" units.
Next, we need to determine how many unit squares are along the edges and diagonals of the square. The edges of the square consist of "s" unit squares on each side, making a total of 4s unit squares along the edges.
The diagonals of the square consist of 2 diagonals, each passing through the center of the square. Each diagonal will intersect odd-numbered unit squares along its path. Since the square has an odd side length of "s", the number of unit squares along each diagonal will be (s-1).
Thus, the total number of unit squares along the diagonals is 2 * (s-1).
To find the number of unit squares remaining in the square, we subtract the total number of unit squares along the edges and diagonals from the total number of unit squares in the square.
The total number of unit squares in the square is equal to the square of the side length. So the total number of squares is s^2.
Finally, to get the fully simplified expression for the number of unit squares remaining, we subtract the sum of the unit squares along the edges and diagonals from the total number of unit squares:
Number of unit squares remaining = s^2 - (4s + 2(s-1))
Now, let's simplify the expression:
Number of unit squares remaining = s^2 - (4s + 2s - 2)
= s^2 - 6s + 2
Therefore, the fully simplified expression for the number of unit squares remaining in terms of "s" is s^2 - 6s + 2.