2016 AMC 10B Problem 10

P1-10~2 minPrint

A thin piece of wood of uniform density in the shape of an equilateral triangle with side length $3$ inches weighs $12$ ounces. A second piece of the same type of wood, with the same thickness, also in the shape of an equilateral triangle, has side length of $5$ inches. Which of the following is closest to the weight, in ounces, of the second piece?

Show solution without answering

Key insight

Same wood and same thickness means weight goes with area. Sides grow 5/3 times, so area grows (5/3)^2 = 25/9 times: 12 * 25/9 is about 33.3.

Solution

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 $\boxed{\textbf{(D)}\ 33.3}$ .

Why this works

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.

Alternative approach

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.

Concise solution

A shorter write-up, the way contest solutions are usually written. It may take a different route from the solution above.

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 $\boxed{\textbf{(D)}\ 33.3}$ .

Review in the AoPS books

Introduction to Geometry, Chapter 3: Congruent Triangles
Introduction to Algebra, Chapter 6: Ratios and Percents
Introduction to Geometry, Chapter 5: Similar Triangles
Introduction to Algebra, Chapter 7: Proportion (§7.1, §7.2, §7.3)