Alex built a snowman using three snowballs, one small, one medium and one large, with diameters in the ratio 2:3:5. Although highly unlikely due to the physical properties of snow, suppose the three snowballs were perfectly spherical and stacked vertically, one on top of the other, as shown, with adjacent snowballs sharing a single point of tangency. If the diameter of the medium snowball was 18 inches, what was the maximum height, in feet, of the snowman Alex built using the three snowballs?

\( \frac{3}{5}=\frac{18}{x} \)

\( \boxed{\boxed{x=30~in}} \)

and

\( \frac{3}{2}=\frac{18}{y} \)

\( y=12~in \)

\( Maximum~~height~~=~~12+18+30~~=~~60~in \)

them

\( 1~in=\frac{1}{12}~ft \)

\( 60~in=5~ft \)

\( \boxed{\boxed{\boxed{\therefore~Maximum~~height~~=~~60~in~~=~~5~ft}}} \)

\( ratio\ snowballs:\ \ 2:3:5\\ medium\ snowball - 18\ in\\ \\ x-large\ snowball\\ \frac35=\frac{18}x\\ 3x=90\\ x=30\ in\\\\ \\ y-small\ snowballs\\ \frac23=\frac y{18}\\ 3y=36\\ y=12\ in\\ \\ 12+30+18=60\ in\\ \\ 1\ in - \frac1{12}\ ft\\ 60\ in - z\ ft\\ \\ z=\frac1{12} \cdot 60= 5\ ft\\ Answer: Maximum\ height\ is\ 5\ ft \)


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