Problem Breakdown and Solutions:

Question 2

• Given: Priya is stacking boxes. Each box is 2.25 cm tall. She starts with 3 boxes.
• a) Complete the table for $h(n)$, the height of the stack after $n$ additional boxes are added.

We calculate $h(n)$: $h(n) = \text{Initial height with 3 boxes} + \text{Height of added boxes}$ The initial height with 3 boxes: $3 \times 2.25 = 6.75 \, \text{cm.}$ For each additional box: $\text{Height added} = n \times 2.25.$ Thus, the function becomes: $h(n) = 6.75 + 2.25n.$

• b) Does $h(8)$ make sense? To calculate $h(8)$: $h(8) = 6.75 + 2.25 \times 8 = 6.75 + 18 = 24.75 \, \text{cm.}$ This makes sense because the formula is valid for any non-negative integer $n$ and represents the height after stacking boxes.

Question 3

• Given: A piece of paper starts with an area of 54 square inches. Each person cuts $\frac{1}{3}$ of the remaining area.

• a) Complete the table for $A(n)$, the area after $n$ cuts.

The recursive pattern is: $A(0) = 54,$ $A(n) = \frac{2}{3} A(n-1).$ Calculate: $A(1) = \frac{2}{3} \times 54 = 36,$ $A(2) = \frac{2}{3} \times 36 = 24,$ $A(3) = \frac{2}{3} \times 24 = 16.$

• b) Define $A(n)$ for the $n$-th term: The explicit formula for $A(n)$: $A(n) = 54 \times \left(\frac{2}{3}\right)^n.$

• c) Reasonable domain: Since the cutting process continues indefinitely, $n$ can be any non-negative integer: $\text{Domain: } n \in \mathbb{Z}_{\geq 0}.$ This makes sense because $n$ represents the number of cuts, and it cannot be negative.

Next Steps

Let me know if you'd like further clarifications!
Here are 5 related questions to extend learning:

1. How would the formula for $h(n)$ change if the initial stack had 5 boxes?
2. What happens to $A(n)$ as $n \to \infty$?
3. How do recursive and explicit formulas differ in practical applications?
4. Can $h(n)$ or $A(n)$ have real (non-integer) values for $n$?
5. How would the stacking formula change if each box had a different height?

Tip: Always look for patterns in repetitive calculations to derive general formulas quickly!