Engineering Your Future
Engineering Your Future
9th Edition
ISBN: 9780190279264
Author: William C. Oakes, Les L. Leone
Publisher: Oxford University Press, USA
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Chapter 7, Problem 7.1EAA
To determine

Determine how many and which lockers will be open at the end of the student’s activities.

Expert Solution & Answer
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Explanation of Solution

First let us consider for 15 lockers and 15 students.

123456789101112131415
1OOOOOOOOOOOOOOO
2OCOCOCOCOCOCOCO
3OCCCOOOCCCOOOCC
4OCCOOOOOCCOCOCC
5OCCOCOOOCOOCOCO
6OCCOCCOOCOOOOCO
7OCCOCCCOCOOOOOO
8OCCOCCCCCOOOOOO
9OCCOCCCCOOOOOOO
10OCCOCCCCOCOOOOO
11OCCOCCCCOCCOOOO
12OCCOCCCCOCCCOOO
13OCCOCCCCOCCCCOO
14OCCOCCCCOCCCCCO
15OCCOCCCCOCCCCCC

Let O = Locker Open.

 C = Locker Closed.

From the above table, we can make out that there are 3 lockers open at the end of 15th student activity.

For 15 lockers and 15 students, the highest of the perfect squares number is 9 that is 3*3(that is, below number 15, we have 1, 4 (22), and 9 (32) as the perfect square numbers. In that, highest is 9). The square root of the highest of the perfect squares number gives the number of lockers open, that is v9 = 3.

Similarly, for 1000 lockers and students, 961 (31*31=961) is the highest of the perfect squares number. Therefore, v961 = 31, that is 31 lockers are open at the end of 1000th student activity.

So, in an easier way we can say, for a particular student and locker number, the square root of highest of the perfect squares number is the number of lockers that will be open.

All the perfect square numbers (1, 4, 9, 16, 25….till 961) are the lockers that will be open.

Similarly, you can try for the below student and locker numbers:

For, 20 lockers and students, the number of lockers open is 4 (since 16 = 4*4 is the highest of the perfect squares number below 20)

For, 30 lockers and students, the number of lockers open is 5 (since 25 = 5*5 is the highest of the perfect squares number below 30)

Conclusion:

There are 31 lockers open at the end of 1000th student activity. Also, all the perfect square numbers (1, 4, 9, 16, 25.....till 961) are the lockers that will be open.

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