A sheet is folded in half a number of times - Junior Cycle Mathematics - Question 6 - 2018

Domain = {$1, 2, 3, 4, 5, 6$} Range = {$2, 4, 8, 16, 32, 64$}

To find the number of folds $x$ such that $h(x) > 500$, we use the equation: $2^x > 500$ Taking logarithms on both sides: $x > \log_2(500)$ Calculating this gives: $x \approx 8.97$ Thus, the least integer greater than 8.97 is 9. Thus, 9 folds are needed.

For 12 folds: $h(12) = 2^{12} = 4096$ We can express this in the form $a \times 10^n$. First, we note that: $4096 = 4.096 \times 10^3$ Thus, $a = 4.096$ and $n = 3$.

$h(14) > 10 000$ means that after 14 folds of the paper, the number of layers exceeds 10,000. This indicates that the exponential growth of layers reaches significant numbers quickly due to repeated doubling.

The pattern formed by the number of layers is exponential because the number of layers doubles with each fold.

Justification: The relationship is modeled by the function $h(x) = 2^x$, which is characteristic of exponential growth.

If there are $k$ layers after a certain number of folds, then the number of layers after 3 more folds will be: $k \times 2^3 = 8k$

If there are $2^p$ layers after a certain number of folds, then the number of layers after 3 more folds will be: $2^p \times 2^3 = 2^{p+3}$