Find the Height of One Cup: Solving the Stacked Cups Problem with 39 cm and 21 cm

To solve this problem, we need to understand how stacked cups work. When cups are stacked, they nest into each other, so each additional cup doesn't add its full height to the stack.

Setting Up Variables:
Let $h$ = the full height of one cup (in cm)
Let $x$ = the amount of overlap when one cup nests into another (in cm)

Understanding the Stacking Pattern:
When cups are stacked:

• The first (bottom) cup contributes its full height: $h$
• Each additional cup only adds $(h - x)$ to the total height, since it overlaps by $x$ cm
• For $n$ cups, the total height is: $h + (n-1)(h-x)$

Setting Up the Equations:
For a stack of 5 cups with height 39 cm:
$h + 4(h - x) = 39$
Simplifying: $h + 4h - 4x = 39$
$5h - 4x = 39$ ... (Equation 1)

For a stack of 2 cups with height 21 cm:
$h + 1(h - x) = 21$
Simplifying: $h + h - x = 21$
$2h - x = 21$ ... (Equation 2)

Solving the System of Equations:
From Equation 2: $x = 2h - 21$

Substituting into Equation 1:
$5h - 4(2h - 21) = 39$
$5h - 8h + 84 = 39$
$-3h + 84 = 39$
$-3h = -45$
$h = 15$

Verification:
If $h = 15$, then $x = 2(15) - 21 = 30 - 21 = 9$ cm
Check with 5 cups: $15 + 4(15 - 9) = 15 + 4(6) = 15 + 24 = 39$ ✓
Check with 2 cups: $15 + 1(15 - 9) = 15 + 6 = 21$ ✓

Therefore, the height of one cup is 15 cm.