2016 AMC 10B Problem 10

Question

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

5

inches. Which of the following is closest to the weight, in ounces, of the second piece?

\ 14.0

\ 16.0

\ 20.0

\ 33.3

\ 55.6

Answer: D

Concepts:similarity power scaling of length, area, and volume

Difficulty rating: 960

Small hint:

Weights scale like areas because thickness and density are fixed

Big hint:

The side-length scale factor is

\frac{5}{3}

Solution:

The surface area is increased by a factor of

\frac 53 ^2

since the side lengths are increased by a factor of

\frac 53.

Also since the thickness is constant, the volume is scaled up by that much. Then, the wood having the same density makes the weight increase by a factor of

\frac 53^2

as well, so the new weight is

12 \cdot \dfrac {25}{9} = \dfrac{100}3.

This is approximately

33.3.

Thus, the correct answer is D .