Question

John is asked to clean a circular radioactive area of radius r. if by the end of each day he cleans half the area that he started with on that day, how much area would be left to clean at the end of the twentieth day?

Answer 1

Initially, the area of the circular radioactive region = πr² = 3.14r²

At the end of the first day, he shall have cleaned half of this area = 1/2x3.14r² = 1.57r²

Since at the end of each day, he cleans half of the area he started with on the that day, the areas cleaned will form a geometric sequence with first term ,
a = 1.57r² and common ratio = 1/2 = 0.5

To find the total area cleaned by the end of the 20th day we use the formula for finding the sum of the geometric sequence terms up to the 20th term.

Sn = a(1-r^n)/1-r

S20 = 1.57r² (1-0.5^20)/(1-0.5)
S20 = 3.139997005r²

The remaining area = 3.14r² - 3.139997005r²

                                = 0.000002995r²