If the sign was nailed to the side of a tree 5' above the ground in 1997. How high would the sign be in 2007 if the tree grew 4" taller each year?

The sign remains at 5'high because the growth of height known as primary growth in the shoot is always happens at the tip of the shoot where apical meristem is present so the new cells form deposit over older cells from the above. Hence height of the sign remains the same.