A number of identical cups are stacked up. The problem asks us to determine: a. If the height of the stack is a function of the number of cups in the stack. b. If the number of cups in a stack is a function of the height of the stack.

1. Problem Description

A number of identical cups are stacked up. The problem asks us to determine:

a. If the height of the stack is a function of the number of cups in the stack.

b. If the number of cups in a stack is a function of the height of the stack.

2. Solution Steps

a. Can we say that the height of the stack is a function of the number of cups in the stack? Explain your reasoning.

Let $n$ be the number of cups in the stack, and $h$ be the height of the stack.

If we know the number of cups $n$, we can determine the height $h$. Specifically, let $h_1$ be the height of one cup and $h_o$ be the overlap in height when cups are stacked (if the cups can nest together). Then, the height of the stack is given by $h = h_1 + (n-1)h_o$, where $h_o < h_1$. Alternatively, if they don't nest at all, $h = n h_1$. In either case, if we know the number of cups, we can uniquely determine the height of the stack.

Therefore, the height of the stack is a function of the number of cups.

b. Can we say that the number of cups in a stack is a function of the height of the stack? Explain your reasoning.

Let $n$ be the number of cups in the stack, and $h$ be the height of the stack.

If we know the height of the stack $h$, we can determine the number of cups $n$. We can rearrange the equations we had before to find the number of cups. Using the first equation:

$h = h_1 + (n-1)h_o$

$h - h_1 = (n-1)h_o$

$\frac{h - h_1}{h_o} = n-1$

$n = \frac{h - h_1}{h_o} + 1$

Using the second equation:

$h = n h_1$

$n = \frac{h}{h_1}$

In either case, if we know the height of the stack, we can uniquely determine the number of cups.

Therefore, the number of cups is a function of the height of the stack.

3. Final Answer

a. Yes, the height of the stack is a function of the number of cups in the stack. For a given number of cups, there is only one possible height of the stack.

b. Yes, the number of cups in a stack is a function of the height of the stack. For a given height of the stack, there is only one possible number of cups in the stack.