2016 AMC 10B Problems/Problem 10
Problem
A thin piece of wood of uniform density in the shape of an equilateral triangle with side length inches weighs 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 inches. Which of the following is closest to the weight, in ounces, of the second piece?
Solution 1
We can solve this problem by using similar triangles, since two equilateral triangles are always similar. We can then use
.
We can then solve the equation to get which is closest to
Solution 2
Also recall that the area of an equilateral triangle is so we can give a ratio as follows:
Cross multiplying and simplifying, we get
Which is
- Solution by
Solution 3
Note that the ratio of the two triangle's weights is equal to the ratio of their areas, as the height is the same. The ratio of their areas is equal to the square of the ratio of their sides. So if denotes the weight of the second triangle, we have Solving gives us so the answer is .
Note: In general it would be volume. However, we are given the that they have equal height, so we essentially treat the problem as 2D.
Video Solution
~savannahsolver
See Also
2016 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 9 |
Followed by Problem 11 | |
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All AMC 10 Problems and Solutions |
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