2014 AMC 12A Problem 17

The four top small-sphere centers form a square of side $2$, directly below the big sphere’s center

With center distance $3$ (sum of radii) and horizontal offset $\sqrt2$, the vertical gap is $\sqrt{3^2-\sqrt2^2}=\sqrt7$; then add $1$ at the top and bottom

Place the box with a corner at the origin. Each small sphere sits in a corner with center $1$ unit from three faces. The four top small-sphere centers form a square of side $2,$ whose center lies on the box axis; a corner of that square is $\sqrt2$ from the center.

The big sphere’s center is on the axis, at distance $2+1=3$ from each top small center. The vertical gap between them is $\sqrt{3^2-\sqrt2^2}=\sqrt7.$

The big center is at height $\dfrac h2$ and the top small centers at height $h-1,$ so $(h-1)-\dfrac h2=\sqrt7,$ giving $\dfrac h2=1+\sqrt7$ and $h=2+2\sqrt7.$

Thus, the correct answer is A.