2016 AMC 10B Problem 10

1. Both pieces are the same wood with the same thickness. So a piece twice as big in area weighs twice as much. Weight follows area.
2. The two triangles have the same shape. The sides grow from $3$ to $5$, which is $\frac53$ times as long.
3. When every length is multiplied by $\frac53$, the area is multiplied by $\frac53 \cdot \frac53 = \frac{25}{9}$. (Think of a square: making both its sides $2$ times as long makes the area $4$ times as big.)
4. So the new weight is
$$12 \cdot \frac{25}{9} = \frac{300}{9} = \frac{100}{3} \approx 33.3 \text{ ounces}.$$

The answer is $\boxed{\textbf{(D)}\ 33.3}$.

For two shapes that are scaled copies of each other, area grows with the square of the length ratio. You never need the area formula for an equilateral triangle; the ratio does all the work. Whenever the same shape appears at two sizes, square the ratio of lengths to compare areas.

Compare triangle counts. A triangle with side $3$ can be cut into $3^2 = 9$ small triangles of side $1$, and one with side $5$ into $5^2 = 25$ of them. So each small triangle weighs $\frac{12}{9}$ ounces, and the big piece weighs $25 \cdot \frac{12}{9} = \frac{100}{3} \approx 33.3$ ounces.

Both pieces have the same density and thickness, so their weights are proportional to their areas.

The two equilateral triangles are similar with side ratio $\frac{5}{3}$, so their areas are in the ratio $\left(\frac{5}{3}\right)^2 = \frac{25}{9}$.

Therefore the second piece weighs
$$12 \cdot \frac{25}{9} = \frac{300}{9} = \frac{100}{3} \approx 33.3 \text{ ounces}.$$

The answer is $\boxed{\textbf{(D)}\ 33.3}$.