A Square and A CubeClass 8 Mathematics Case Study Questions

A neighborhood is planning a new community garden on a square plot of land with an area of $729$ square meters. The residents plan to put a fence around the entire perimeter of the garden. For irrigation, they will install a large cubic water tank. The chosen tank has a volume of $216$ cubic meters. The cost of fencing material is $₹50$ per meter. The garden will be further divided into 9 equal smaller square patches for different types of vegetables.

For the annual day function, a school needs to arrange chairs for students in the main ground. The event manager decides to arrange the chairs in a square formation, with an equal number of rows and columns. There are a total of $1156$ chairs available. After the arrangement, the remaining chairs are to be stacked. The school also has a collection of $1331$ prize boxes, which are identical cubes, to be arranged in a large cubic display for the prize distribution ceremony.

A new school, "Aryabhata Academy", designs a fun orientation activity for its 150 new students. There are 150 lockers, numbered 1 to 150, all initially closed. Student 1 opens every locker. Student 2 toggles every 2nd locker (closes it). Student 3 toggles every 3rd locker, and so on, until student 150 toggles the 150th locker. At the end of the process, a locker remains open only if it was toggled an odd number of times. This happens for lockers whose numbers have an odd number of factors. The final passcode to the school's library is formed by the first three 2-digit numbers of the lockers that remain open.