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Answer :
One would need to fold the paper approximately 11 times to reach the height of the Empire State Building. Exponential functions are necessary to solve this problem due to the exponential growth pattern of the paper's thickness with each fold, which cannot be accurately represented by simple addition.
Exponential functions are needed to solve this problem because each fold of the paper doubles its thickness. When we fold the paper, the resulting thickness is not simply additive but follows an exponential growth pattern. The thickness of the paper after each fold can be represented as [tex]0.003 ft * 2^n,[/tex] where n is the number of folds.
To determine the number of folds needed to reach the height of the Empire State Building (1450 ft), we can set up the equation:
[tex]0.003 ft * 2^n = 1450 ft[/tex]
By solving this exponential equation, we find that n is approximately equal to 11. This means that the paper needs to be folded 11 times to reach a thickness of 1450 ft, equivalent to the height of the Empire State Building.
Exponential functions are crucial in this context as they describe the rapid and compounded growth of the paper's thickness with each fold, allowing us to determine the number of folds required to reach a specific height.
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