Grades 9-10 Video Solutions 2025-2025_9-10

Video Transcription

Problem 23. Daniel numbers certain squares on a sheet of grid paper. Each square has a side length of 0.5 centimeters. He starts with the first square and then numbers the  squares 2, 3, 4, 5 in a counterclockwise direction as shown. He stops when he has numbered 2,025  squares and looks at the figure made of all the number of squares. What is the perimeter of this figure? So first observe that if n is a square number and Daniel has numbered  n squares, then the spiral that Daniel has drawn is going to be a square of n by n boxes. So for example, the first square number is 1, and after 1 box, this is a square with  1 box, right? The next square number is 4, and after Daniel has numbered 4 squares, this is going to be the shape. This is a 2 by 2 square. Then this is the square numbered  7, this will be the square numbered 8, and this will be the square numbered 9. So these 9 squares here form a 3 by 3 square. Now we know that 2,025 is 45 squared, and so that  means that after Daniel has numbered 2,025 squares, this figure is going to be a 45 by 45 square. We know that each of the small squares has a side length of 0.5 cm, so that  means the side length of this 45 by 45 square is going to be 45 times 0.5. Then the perimeter is going to be 45 times 0.5 times 4, which is 90 cm, and the answer is D.

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