Question

Please help.This is part C:c A penny is about 0.06 inch thick. About how tall is the tallest stack?

Answer 1

a. (1, 4), (2, 3), (3, 2) and (4, 1)

b. $655.36

c. 3932.16 inches

Step-by-step 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:

`n=2^x\cdot2^y`

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:

`32=2^x\cdot2^y`

Write 32 in form of powers of 2s, we have that:

`\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}`

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:

`2^8\cdot2^8`

This is because 8 is the highest number of rows and columns.

Therefore, the number of pennies in that position is:

`\begin{gathered} n=2^{8\text{ + 8}}=2^(16) \\ n\text{ = 65536 pennies} \end{gathered}`

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