ANSWER
a. (1, 4), (2, 3), (3, 2) and (4, 1)
b. $655.36
c. 3932.16 inches
EXPLANATION
The row numbers are y and the column numbers are x.
Each position contains a stack of pennies and the number of pennies in each stack is given as:
[tex]n=2^x\cdot2^y[/tex]a. We want to find the locations that have 32 as the number of pennies in the stack.
For a stack to contain 32 pennies, it means that:
[tex]32=2^x\cdot2^y[/tex]Write 32 in form of powers of 2s, we have that:
[tex]\begin{gathered} 2^5=2^x\cdot2^y=2^{x\text{ + y}}\text{ (addition law of indices)} \\ A\text{ccording to the law of indices:} \\ \Rightarrow\text{ 5 = }x\text{ + y} \end{gathered}[/tex]This implies that whichever location must contain 32 pennies must have row and column number which add up to 5.
The only combinations that allow for that are:
1 + 4 = 5
2 + 3 = 5
3 + 2 = 5
4 + 1 = 5
Therefore, the coordinates are (in form of (x, y)):
(1, 4), (2, 3), (3, 2) and (4, 1)
b. The tallest stack of pennies will be the one in the position:
[tex]2^8\cdot2^8[/tex]This is because 8 is the highest number of rows and columns.
Therefore, the number of pennies in that position is:
[tex]\begin{gathered} n=2^{8\text{ + 8}}=2^{16} \\ n\text{ = 65536 pennies} \end{gathered}[/tex]Each penny costs $0.01.
Therefore, the amount of money in the tallest stack is:
Amount = 65536 * 0.01
A = $655.36
c. A penny is about 0.06 inch thick.
Since there are 65536 pennies in the tallest stack, it means that, the height of the tallest stack is:
H = 65536 * 0.06
H = 3932.16 inches